Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using...
Transcript of Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using...
![Page 1: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/1.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 1 GEOMETRY
Lesson 1: Scale Drawings Date: 9/26/14 S.1
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Lesson 1: Scale Drawings
Classwork
Opening Exercise
Above is a picture of a bicycle. Which of the images below appears to be a well-scaled image of the original? Why?
![Page 2: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/2.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 1 GEOMETRY
Lesson 1: Scale Drawings Date: 9/26/14 S.2
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Example 1
Use construction tools to create a scale drawing of โณ ๐ด๐ต๐ถ with a scale factor of ๐ = 2.
Exercise 1
Use construction tools to create a scale drawing of โณ ๐ท๐ธ๐น with a scale factor of ๐ = 3. What properties does your scale drawing share with the original figure? Explain how you know.
![Page 3: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/3.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 1 GEOMETRY
Lesson 1: Scale Drawings Date: 9/26/14 S.3
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Example 2
Use construction tools to create a scale drawing of โณ ๐๐๐ with a scale factor of ๐ = 12.
Exercises 2โ4
2. Use construction tools to create a scale drawing of โณ ๐๐๐ with a scale factor of ๐ = 14. What properties do the scale
drawing and the original figure share? Explain how you know.
![Page 4: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/4.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 1 GEOMETRY
Lesson 1: Scale Drawings Date: 9/26/14 S.4
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3. Triangle ๐ธ๐น๐บ is provided below, and one angle of scale drawing โณ ๐ธโฒ๐นโฒ๐บโฒ is also provided. Use construction tools tocomplete the scale drawing so that the scale factor is ๐ = 3. What properties do the scale drawing and the originalfigure share? Explain how you know.
4. Triangle ๐ด๐ต๐ถ is provided below, and one side of scale drawing โณ ๐ดโฒ๐ตโฒ๐ถโฒ is also provided. Use construction tools tocomplete the scale drawing and determine the scale factor.
![Page 5: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/5.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 1 GEOMETRY
Lesson 1: Scale Drawings Date: 9/26/14 S.5
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Problem Set
1. Use construction tools to create a scale drawing of โณ ๐ด๐ต๐ถ with a scale factor of ๐ = 3.
2. Use construction tools to create a scale drawing of โณ ๐ด๐ต๐ถ with a scale factor of ๐ = 12.
Lesson Summary
There are two properties of a scale drawing of a figure: corresponding angles are equal in measurement, and corresponding lengths are proportional in measurement.
![Page 6: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/6.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 1 GEOMETRY
Lesson 1: Scale Drawings Date: 9/26/14 S.6
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3. Triangle ๐ธ๐น๐บ is provided below, and one angle of scale drawing โณ ๐ธโฒ๐นโฒ๐บโฒ is also provided. Use construction tools tocomplete a scale drawing so that the scale factor is ๐ = 2.
4. Triangle ๐๐๐ถ is provided below, and one angle of scale drawing โณ๐โฒ๐โฒ๐ถโฒ is also provided. Use construction toolsto complete a scale drawing so that the scale factor is 1
4.
![Page 7: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/7.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 1 GEOMETRY
Lesson 1: Scale Drawings Date: 9/26/14 S.7
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5. Triangle ๐ด๐ต๐ถ is provided below, and one side of scale drawing โณ ๐ดโฒ๐ตโฒ๐ถโฒ is also provided. Use construction tools tocomplete the scale drawing and determine the scale factor.
6. Triangle ๐๐๐ is provided below, and one side of scale drawing โณ ๐โฒ๐โฒ๐โฒ is also provided. Use construction tools tocomplete the scale drawing and determine the scale factor.
7. Quadrilateral ๐บ๐ป๐ผ๐ฝ is a scale drawing of quadrilateral ๐ด๐ต๐ถ๐ท with scale factor ๐. Describe each of the followingstatements as always true, sometimes true, or never true, and justify your answer.a. ๐ด๐ต = ๐บ๐ปb. ๐โ ๐ด๐ต๐ถ = ๐โ ๐บ๐ป๐ผ
c. ๐ด๐ต๐บ๐ป
= ๐ต๐ถ๐ป๐ผ
d. Perimeter(๐บ๐ป๐ผ๐ฝ) = ๐ โ Perimeter(๐ด๐ต๐ถ๐ท)e. Area(๐บ๐ป๐ผ๐ฝ) = ๐ โ Area(๐ด๐ต๐ถ๐ท) where ๐ โ 1f. ๐ < 0
![Page 8: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/8.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 GEOMETRY
Lesson 2: Making Scale Drawings Using the Ratio Method Date: 9/26/14 S.8
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Lesson 2: Making Scale Drawings Using the Ratio Method
Classwork
Opening Exercise
Based on what you recall from Grade 8, describe what a dilation is.
Example 1
Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐ = 12.
Step 1. Draw a ray beginning at ๐ through each vertex of the figure.
Step 2. Dilate each vertex along the appropriate ray by scale factor ๐ = 12. Use the ruler to find the midpoint between ๐
and ๐ท and then each of the other vertices. Label each respective midpoint with prime notation, i.e., ๐ทโ.
Step 3. Join vertices in the way they are joined in the original figure, e.g., segment ๐ดโฒ๐ตโฒ corresponds to segment ๐ด๐ต.
![Page 9: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/9.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 GEOMETRY
Lesson 2: Making Scale Drawings Using the Ratio Method Date: 9/26/14 S.9
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exercise 1
1. Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐ = 34. Verify that
the resulting figure is in fact a scale drawing by showing that corresponding side lengths are in constant proportionand the corresponding angles are equal in measurement.
Example 2
a. Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐ = 3.
Step 1. Draw a ray beginning at ๐ through each vertex of the figure.
Step 2. Use your ruler to determine the location of ๐ดโฒ on ๐๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ ; ๐ดโฒ should be three times as far from ๐ as ๐ด.Determine the locations of ๐ตโฒ and ๐ถโฒ in the same way along the respective rays.
Step 3. Draw the corresponding line segments, e.g., segment ๐ดโฒ๐ตโฒ corresponds to segment ๐ด๐ต.
![Page 10: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/10.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 GEOMETRY
Lesson 2: Making Scale Drawings Using the Ratio Method Date: 9/26/14 S.10
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b. Locate a point ๐ so that it lies between endpoints ๐ด and ๐ต on segment ๐ด๐ต of the original figure in part (a). Use theratio method to locate ๐โฒ on the scale drawing in part (a).
c. Imagine a dilation of the same figure as in parts (a) and (b). What if the ray from the center passed through twodistinct points, such as ๐ต and ๐ท below? What does that imply about the locations of ๐ตโฒ and ๐ทโฒ?
Exercises 2โ6
2. โณ ๐ดโฒ๐ตโฒ๐ถโฒ is a scale drawing of โณ ๐ด๐ต๐ถ drawn by using the ratio method. Use your ruler to determine the location ofthe center ๐ used for the scale drawing.
![Page 11: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/11.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 GEOMETRY
Lesson 2: Making Scale Drawings Using the Ratio Method Date: 9/26/14 S.11
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3. Use the figure below with center ๐ and a scale factor of ๐ = 52 to create a scale drawing. Verify that the resulting
figure is in fact a scale drawing by showing that corresponding side lengths are in constant proportion and that thecorresponding angles are equal in measurement.
4. Summarize the steps to create a scale drawing by the ratio method. Be sure to describe all necessary parameters touse the ratio method.
5. A clothing company wants to print the face of the Statue of Liberty on a T-shirt. The length of the face from the topof the forehead to the chin is 17 feet and the width of the face is 10 ft. Given that a medium sized T-shirt has alength of 29 in and a width of 20 in, what dimensions of the face are needed to produce a scaled version that will fiton the T-shirt?
a. What shape would you use to model the face of the statue?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 GEOMETRY
Lesson 2: Making Scale Drawings Using the Ratio Method Date: 9/26/14 S.12
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b. Knowing that the maximum width of the T-shirt is 20 in, what scale factor is needed to make the width of theface fit on the shirt?
c. What scale factor should be used to scale the length of the face? Explain.
d. Using the scale factor identified in part (c), what is the scaled length of the face? Will it fit on the shirt?
e. Identify the scale factor you would use to ensure that the face of the statue was in proportion and would fit onthe T-shirt. Identify the dimensions of the face that will be printed on the shirt.
f. The T-shirt company wants the width of the face to be no smaller than 10 inches. What scale factors could beused to create a scaled version of the face that meets this requirement?
g. If it costs the company $0.005 for each square inch of print on a shirt, what is the maximum and minimumcosts for printing the face of the statue of liberty on one T-shirt?
![Page 13: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/13.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 GEOMETRY
Lesson 2: Making Scale Drawings Using the Ratio Method Date: 9/26/14 S.13
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6. Create your own scale drawing using the ratio method. In the space below:
a. Draw an original figure.
b. Locate and label a center of dilation ๐.c. Choose a scale factor ๐.
d. Describe your dilation using appropriate notation.
e. Complete a scale drawing using the ratio method.
Show all measurements and calculations to confirm that the new figure is a scale drawing. The work here will be your answer key. Next, trace your original figure onto a fresh piece of paper. Trade the traced figure with a partner. Provide your partner with the dilation information. Each partner should complete the otherโs scale drawing. When finished, check all work for accuracy against your answer key.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 GEOMETRY
Lesson 2: Making Scale Drawings Using the Ratio Method Date: 9/26/14 S.14
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Problem Set
1. Use the ratio method to create a scale drawing about center ๐ with a scale factor of ๐ = 14. Use a ruler and
protractor to verify that the resulting figure is in fact a scale drawing by showing that corresponding side lengths arein constant proportion and the corresponding angles are equal in measurement.
2. Use the ratio method to create a scale drawing about center ๐ with a scale factor of ๐ = 2. Verify that the resultingfigure is in fact a scale drawing by showing that corresponding side lengths are in constant proportion and that thecorresponding angles are equal in measurement.
Lesson Summary
1. To create a scale drawing using the ratio method, each vertex of the original figure is dilated about the center ๐by scale factor ๐. Once all the vertices are dilated, they are joined to each other in the same way as in theoriginal figure.
2. The scale factor tells us whether the scale drawing is being enlarged (๐ > 1) or reduced (0 < ๐ < 1).
![Page 15: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/15.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 GEOMETRY
Lesson 2: Making Scale Drawings Using the Ratio Method Date: 9/26/14 S.15
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
3. Use the ratio method to create two scale drawings: ๐ท๐,2 and ๐ท๐,2. Label the scale drawing with respect to center ๐as โณ ๐ดโฒ๐ตโฒ๐ถโฒ and the scale drawing with respect to center ๐ as โณ ๐ดโฒโฒ๐ตโฒโฒ๐ถโฒโฒ.
What do you notice about the two scale drawings?
What rigid motion can be used to map โณ ๐ดโฒ๐ตโฒ๐ถโฒ onto โณ ๐ดโฒโฒ๐ตโฒโฒ๐ถโฒโฒ?
4. Sara found a drawing of a triangle that appears to be a scale drawing. Much of the drawing has faded, but she cansee the drawing and construction lines in the diagram below. If we assume the ratio method was used to constructโณ ๐ดโฒ๐ตโฒ๐ถโฒ as a scale model of โณ ๐ด๐ต๐ถ, can you find the center ๐, the scale factor ๐, and locate โณ ๐ด๐ต๐ถ?
![Page 16: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/16.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 2 GEOMETRY
Lesson 2: Making Scale Drawings Using the Ratio Method Date: 9/26/14 S.16
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5. Quadrilateral ๐ดโฒโฒโฒ๐ตโฒโฒโฒ๐ถโฒโฒโฒ๐ทโฒโฒโฒ is one of a sequence of three scale drawings of quadrilateral ๐ด๐ต๐ถ๐ท that were allconstructed using the ratio method from center ๐. Find the center ๐, each scale drawing in the sequence and thescale factor for each scale drawing. The other scale drawings are quadrilaterals ๐ดโฒ๐ตโฒ๐ถโฒ๐ทโฒ and ๐ดโฒโฒ๐ตโฒโฒ๐ถโฒโฒ๐ทโฒโฒ.
6. Maggie has a rectangle drawn in the corner of a 8 12 inch by 11 inch sheet of printer paper as shown in the diagram.
To cut out the rectangle, Maggie must make two cuts. She wants to scale the rectangle so that she can cut it outusing only one cut with a paper cutter.
a. What are the dimensions of Maggieโs scaled rectangleand what is its scale factor from the original rectangle?
b. After making the cut for the scaled rectangle, is thereenough material left to cut another identical rectangle?If so, what is the area of scrap per sheet of paper?
![Page 17: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/17.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 3 GEOMETRY
Lesson 3: Making Scale Drawings Using the Parallel Method Date: 9/26/14 S.17
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Lesson 3: Making Scale Drawings Using the Parallel Method
Classwork
Opening Exercise
Dani dilated โณ ๐ด๐ต๐ถ from center ๐, resulting in โณ ๐ดโฒ๐ตโฒ๐ถโฒ. She says that she completed the drawing using parallel lines. How could she have done this? Explain.
Example 1
a. Use a ruler and setsquare to draw a line through ๐ถ parallel to ๐ด๐ต. What ensures that the line drawn is parallelto ๐ด๐ต?
![Page 18: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/18.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 3 GEOMETRY
Lesson 3: Making Scale Drawings Using the Parallel Method Date: 9/26/14 S.18
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b. Use a ruler and setsquare to draw a parallelogram ๐ด๐ต๐ถ๐ท around ๐ด๐ต and point ๐ถ.
Example 2
Use the figure below with center ๐ and a scale factor of ๐ = 2 and the following steps to create a scale drawing using the parallel method.
Step 1. Draw a ray beginning at ๐ through each vertex of the figure.
Step 2. Select one vertex of the scale drawing to locate; we have selected ๐ดโฒ. Locate ๐ดโฒ on ray ๐๐ด ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ so that ๐๐ดโฒ = 2๐๐ด.
Step 3. Align the setsquare and ruler as in the image below; one leg of the setsquare should line up with side ๐ด๐ต, and the perpendicular leg should be flush against the ruler.
Step 4. Slide the setsquare along the ruler until the edge of the setsquare passes through ๐ดโฒ. Then, along the perpendicular leg of the setsquare, draw the segment through ๐ดโฒ that is parallel to ๐ด๐ต until it intersects with ๐๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ , and label this point ๐ตโฒ.
Step 5. Continue to create parallel segments to determine each successive vertex point. In this particular case, the setsquare has been aligned with ๐ด๐ถ. This is done because, in trying to create a parallel segment from ๐ต๐ถ, the parallel segment was not โreachingโ ๐ตโฒ. This could be remedied with a larger setsquare and longer ruler, but it is easily avoided by working on the segment parallel to ๐ด๐ถ instead.
Step 6. Use your ruler to join the final two unconnected vertices.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 3 GEOMETRY
Lesson 3: Making Scale Drawings Using the Parallel Method Date: 9/26/14 S.19
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Exercises 1โ3
1. With a ruler and setsquare, use the parallel method to create a scale drawing of ๐๐๐๐ by the parallel method. ๐โฒhas already been located for you. Determine the scale factor of the scale drawing. Verify that the resulting figure isin fact a scale drawing by showing that corresponding side lengths are in constant proportion and thatcorresponding angles are equal in measurement.
2. With a ruler and setsquare, use the parallel method to create a scale drawing of ๐ท๐ธ๐น๐บ about center ๐ with scalefactor ๐ = 1
2. Verify that the resulting figure is in fact a scale drawing by showing that corresponding side lengths are
in constant proportion and that the corresponding angles are equal in measurement.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 3 GEOMETRY
Lesson 3: Making Scale Drawings Using the Parallel Method Date: 9/26/14 S.20
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3. With a ruler and setsquare, use the parallel method to create a scale drawing of pentagon ๐๐๐ ๐๐ about center ๐with scale factor 5
2. Verify that the resulting figure is in fact a scale drawing by showing that corresponding side
lengths are in constant proportion and that corresponding angles are equal in measurement.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 3 GEOMETRY
Lesson 3: Making Scale Drawings Using the Parallel Method Date: 9/26/14 S.21
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Problem Set
1. With a ruler and setsquare, use the parallel method to create a scale drawing of the figure about center ๐. Onevertex of the scale drawing has been provided for you.
Determine the scale factor. Verify that the resulting figure is in fact a scale drawing by showing that corresponding side lengths are in constant proportion and that the corresponding angles are equal in measurement.
2. Determine the scale factor. Verify that the resulting figure is in fact a scale drawing by showing that correspondingside lengths are in constant proportion and that the corresponding angles are equal in measurement.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 3 GEOMETRY
Lesson 3: Making Scale Drawings Using the Parallel Method Date: 9/26/14 S.22
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3. With a ruler and setsquare, use the parallel method to create the following scale drawings about center ๐: (1) firstuse a scale a factor of 2 to create โณ ๐ดโฒ๐ตโฒ๐ถโฒ, (2) then, with respect to โณ ๐ดโฒ๐ตโฒ๐ถโฒ, use a scale factor of 2
3 to create scale
drawing โณ ๐ด"๐ต"๐ถ". Calculate the scale factor for โณ ๐ดโฒโฒ๐ตโฒโฒ๐ถโฒโฒ as a scale drawing of โณ ๐ด๐ต๐ถ. Use angle and side lengthmeasurements and the appropriate proportions to verify your answer.
4. Follow the direction in each part below to create three scale drawings of โณ ๐ด๐ต๐ถ using the parallel method.
a. With the center at vertex ๐ด, make a scale drawing of โณ ๐ด๐ต๐ถ with a scale factor of 32.
b. With the center at vertex ๐ต, make a scale drawing of โณ ๐ด๐ต๐ถ with a scale factor of 32.
c. With the center at vertex ๐ถ, make a scale drawing of โณ ๐ด๐ต๐ถ with a scale factor of 32.
d. What conclusions can be drawn about all three scale drawings from parts (a)โ(c)?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 3 GEOMETRY
Lesson 3: Making Scale Drawings Using the Parallel Method Date: 9/26/14 S.23
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5. Use the parallel method to make a scale drawing of the line segments in the following figure using the given ๐โฒ, theimage of vertex ๐, from center ๐. Determine the scale factor.
Use your diagram from Problem 1 to answer this question.
6. If we switch perspective and consider the original drawing ๐ด๐ต๐ถ๐ท๐ธ to be a scale drawing of the constructed image๐ดโฒ๐ตโฒ๐ถโฒ๐ทโฒ๐ธโฒ, what would the scale factor be?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 4 GEOMETRY
Lesson 4: Comparing the Ratio Method with the Parallel Method Date: 9/26/14 S.24
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Lesson 4: Comparing the Ratio Method with the Parallel Method
Classwork Today, our goal is to show that the parallel method and the ratio method are equivalent; that is, given a figure in the plane and a scale factor ๐ > 0, the scale drawing produced by the parallel method is congruent to the scale drawing produced by the ratio method. We start with two easy exercises about the areas of two triangles whose bases lie on the same line, which will help show that the two methods are equivalent.
Opening Exercises 1โ2
1. Suppose two triangles, โณ ๐ด๐ต๐ถ and โณ ๐ด๐ต๐ท, share the same base ๐ด๐ต such that points ๐ถ and ๐ท lie on a line parallel toline ๐ด๐ต๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ . Show that their areas are equal, i.e., Area(โณ ๐ด๐ต๐ถ) = Area(โณ ๐ด๐ต๐ท). (Hint: Why are the altitudes of eachtriangle equal in length?)
2. Suppose two triangles have different length bases, ๐ด๐ต and ๐ด๐ตโฒ, that lie on the same line. Furthermore, supposethey both have the same vertex ๐ถ opposite these bases. Show that value of the ratio of their areas is equal to thevalue of the ratio of the lengths of their bases, i.e.,
Area(โณ๐ด๐ต๐ถ)Area(โณ๐ด๐ตโฒ๐ถ)
= ๐ด๐ต๐ด๐ตโฒ
.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 4 GEOMETRY
Lesson 4: Comparing the Ratio Method with the Parallel Method Date: 9/26/14 S.25
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Discussion
To show that the parallel and ratio methods are equivalent, we need only look at one of the simplest versions of a scale drawing: scaling segments. First, we need to show that the scale drawing of a segment generated by the parallel method is the same segment that the ratio method would have generated and vice versa. (i.e., that the scaled segment generated by the ratio method is the same segment generated by the parallel method.) That is,
The parallel method โน The ratio method,
and
The ratio method โน The parallel method.
The first implication above can be stated as the following theorem:
PARALLEL โน RATIO THEOREM: Given a line segment ๐ด๐ต and point ๐ not on the line ๐ด๐ต๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ , construct a scale drawing of ๐ด๐ต with scale factor ๐ > 0 using the parallel method: Let ๐ดโฒ = ๐ท๐,๐(๐ด), and โ be the line parallel to ๐ด๐ต๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ that passes through ๐ดโฒ. Let ๐ตโฒ be the point where ray ๐๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ intersects โ. Then ๐ตโฒ is the same point found by the ratio method; that is, ๐ตโฒ = ๐ท๐,๐(๐ต).
PROOF: We prove the case when ๐ > 1; the case when 0 < ๐ < 1 is the same but with a different picture. Construct two line segments ๐ต๐ดโฒ and ๐ด๐ตโฒ to form two triangles โณ ๐ต๐ด๐ตโฒ and โณ ๐ต๐ด๐ดโฒ, labeled as ๐1 and ๐2, respectively, in the picture below.
The areas of these two triangles are equal,
Area(๐1) = Area(๐2),
by Exercise 1 (why?). Label โณ ๐๐ด๐ต by ๐0. Then Area(โณ ๐๐ดโฒ๐ต) = Area(โณ ๐๐ตโฒ๐ด) because areas add:
Area(โณ ๐๐ดโฒ๐ต) = Area(๐0) + Area(๐2 )
= Area(๐0) + Area(๐1)
= Area(โณ ๐๐ตโฒ๐ด).
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 4 GEOMETRY
Lesson 4: Comparing the Ratio Method with the Parallel Method Date: 9/26/14 S.26
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Next, we apply Exercise 2 to two sets of triangles: (1) ๐0 and โณ ๐๐ดโฒ๐ต and (2) ๐0 and โณ ๐๐ตโฒ๐ด.
Therefore,
Area(โณ ๐๐ดโฒ๐ต)Area(๐0)
=๐๐ดโฒ
๐๐ด, and
Area(โณ ๐๐ตโฒ๐ด)Area(๐0)
=๐๐ตโฒ
๐๐ต.
Since Area(โณ ๐๐ดโฒ๐ต) = Area(โณ ๐๐ตโฒ๐ด), we can equate the fractions: ๐๐ดโฒ
๐๐ด= ๐๐ตโฒ
๐๐ต. Since ๐ is the scale factor used in
dilating ๐๐ด to ๐๐ดโฒ, we know that ๐๐ดโฒ
๐๐ด= ๐; therefore, ๐๐ต
โฒ
๐๐ต= ๐, or ๐๐ตโฒ = ๐ โ ๐๐ต. This last equality implies that ๐ตโฒ is the
dilation of ๐ต from ๐ by scale factor ๐, which is what we wanted to prove.
Next, we prove the reverse implication to show that both methods are equivalent to each other.
RATIO โน PARALLEL THEOREM: Given a line segment ๐ด๐ต and point ๐ not on the line ๐ด๐ต๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ , construct a scale drawing ๐ดโฒ๐ตโฒ of ๐ด๐ต with scale factor ๐ > 0 using the ratio method (Find ๐ดโฒ = ๐ท๐,๐(๐ด) and ๐ตโฒ = ๐ท๐,๐(๐ต), and draw ๐ดโฒ๐ตโฒ). Then ๐ตโฒ is the same as the point found using the parallel method.
PROOF: Since both the ratio method and the parallel method start with the same first step of setting ๐ดโฒ = ๐ท๐,๐(๐ด), the only difference between the two methods is in how the second point is found. If we use the parallel method, we construct the line โ parallel to ๐ด๐ต๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ that passes through ๐ดโฒ and label the point where โ intersects ๐๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ by ๐ถ. Then ๐ตโฒ is the same as the point found using the parallel method if we can show that ๐ถ = ๐ตโฒ.
(1) ๐ป๐ป๐๐ and โณ๐ถ๐ถ๐จ๐จโฒ๐ฉ๐ฉ with
bases on ๐ถ๐ถ๐จ๐จโฒ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ
(2) ๐ป๐ป๐๐ and โณ๐ถ๐ถ๐ฉ๐ฉโฒ๐จ๐จ with
bases on ๐ถ๐ถ๐ฉ๐ฉโฒ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 4 GEOMETRY
Lesson 4: Comparing the Ratio Method with the Parallel Method Date: 9/26/14 S.27
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By the parallel โน ratio theorem, we know that ๐ถ = ๐ท๐,๐(๐ต), i.e., that ๐ถ is the point on ray ๐๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ such that ๐๐ถ = ๐ โ ๐๐ต. But ๐ตโฒ is also the point on ray ๐๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ such that ๐๐ตโฒ = ๐ โ ๐๐ต. Hence, they must be the same point.
The fact that the ratio and parallel methods are equivalent is often stated as the triangle side splitter theorem. To understand the triangle side splitter theorem, we need a definition:
SIDE SPLITTER: A line segment ๐ถ๐ท is said to split the sides of โณ ๐๐ด๐ต proportionally if ๐ถ is a point on ๐๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, ๐ท is a point on ๐๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, and ๐๐ด
๐๐ถ= ๐๐ต
๐๐ท (or
equivalently, ๐๐ถ๐๐ด
= ๐๐ท๐๐ต
). We call line segment ๐ถ๐ท a side splitter.
TRIANGLE SIDE SPLITTER THEOREM: A line segment splits two sides of a triangle proportionally if and only if it is parallel to the third side.
Restatement of the triangle side splitter theorem:
๐๐ โ ๐ถ๐ถ๐จ๐จ
๐ถ๐ถ๐จ๐จ
๐๐ โ ๐ถ๐ถ๐จ๐จ
๐ถ๐ถ๐จ๐จ
The ratio method The parallel method
C
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 4 GEOMETRY
Lesson 4: Comparing the Ratio Method with the Parallel Method Date: 9/26/14 S.28
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l
Problem Set
1. Use the diagram to answer each part below.
a. Measure the segments in the figure below to verify that the proportion is true.๐๐ดโฒ
๐๐ด=๐๐ตโฒ
๐๐ตb. Is the proportion ๐๐ด
๐๐ดโฒ= ๐๐ต
๐๐ตโฒ also true? Explain algebraically.
c. Is the proportion ๐ด๐ดโฒ
๐๐ดโฒ= ๐ต๐ตโฒ
๐๐ตโฒ also true? Explain algebraically.
2. Given the diagram below, ๐ด๐ต = 30, line โ is parallel to ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, and the distance from ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to โ is 25. Locate point ๐ถ online โ such that โณ ๐ด๐ต๐ถ has the greatest area. Defend your answer.
3. Given โณ ๐๐๐, ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ are partitioned into equal length segments by the endpoints of the dashed segments asshown. What can be concluded about the diagram?
Lesson Summary
THE TRIANGLE SIDE SPLITTER THEOREM: A line segment splits two sides of a triangle proportionally if and only if it is parallel to the third side.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 4 GEOMETRY
Lesson 4: Comparing the Ratio Method with the Parallel Method Date: 9/26/14 S.29
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4. Given the diagram, ๐ด๐ถ = 12, ๐ด๐ต = 6, ๐ต๐ธ = 4, โ ๐ด๐ถ๐ต = ๐ฅยฐ, and โ ๐ท = ๐ฅยฐ, find ๐ถ๐ท.
5. What conclusions can be drawn from the diagram shown to the right? Explain.
6. Parallelogram ๐๐๐ ๐ is shown. Two triangles are formed by a diagonal within the parallelogram. Identify thosetriangles and explain why they are guaranteed to have the same areas.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 4 GEOMETRY
Lesson 4: Comparing the Ratio Method with the Parallel Method Date: 9/26/14 S.30
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7. In the diagram to the right, ๐ป๐ผ = 36 and ๐บ๐ฝ = 42. If the ratio of the areas of the triangles is Area โ๐บ๐ป๐ผArea โ๐ฝ๐ป๐ผ
= 59, find ๐ฝ๐ป,
๐บ๐ป, ๐บ๐ผ, and ๐ฝ๐ผ.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 5 GEOMETRY
Lesson 5: Scale Factors Date: 9/26/14 S.31
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Lesson 5: Scale Factors
Classwork
Opening Exercise
Quick Write: Describe how a figure is transformed under a dilation with a scale factor ๐ = 1, ๐ > 1, and 0 < ๐ < 1.
Discussion
DILATION THEOREM: If a dilation with center ๐ and scale factor ๐ sends point ๐ to ๐โฒ and ๐ to ๐โฒ, then |๐โฒ๐โฒ| = ๐|๐๐|. Furthermore, if ๐ โ 1 and ๐,๐, and ๐ are the vertices of a triangle, then ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ ||๐โฒ๐โฒ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ .
Now consider the dilation theorem when ๐, ๐, and ๐ are the vertices of โณ ๐๐๐. Since ๐โฒ and ๐โฒ come from a dilation
with scale factor ๐ and center ๐, we have ๐๐โฒ
๐๐= ๐๐โฒ
๐๐= ๐.
There are two cases that arise, recall what you wrote in your Quick Write. We must consider the case when ๐ > 1 and when 0 < ๐ < 1. Letโs begin with the latter.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 5 GEOMETRY
Lesson 5: Scale Factors Date: 9/26/14 S.32
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Dilation Theorem Proof, Case 1
Statements Reasons/Explanations
1. A dilation with center ๐ and scale factor r sends point ๐ to๐โฒ and ๐ to ๐โฒ.
2. ๐๐โฒ๐๐
= ๐๐โฒ๐๐
= ๐
3. ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ โ๐โฒ๐โฒ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ
4. A dilation with center ๐ and scale factor ๐๐โฒ
๐๐ sends point ๐
to ๐โฒ and point ๐ to ๐ . Draw ๐โฒ๐ ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
1.
2.
3.
4.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 5 GEOMETRY
Lesson 5: Scale Factors Date: 9/26/14 S.33
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5. ๐โฒ๐ ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ||๐๐โฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
6. ๐ ๐โฒ๐โฒ๐ is a parallelogram.
7. ๐ ๐ = ๐โฒ๐โฒ
8. ๐ ๐๐๐
= ๐โฒ๐๐๐
9. ๐ ๐๐๐
= ๐
10. ๐ ๐ = ๐ โ ๐๐
11. ๐โฒ๐โฒ = ๐ โ ๐๐
5.
6.
7.
8.
9.
10.
11.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 5 GEOMETRY
Lesson 5: Scale Factors Date: 9/26/14 S.34
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Exercises 1โ4
1. Prove Case 2: If ๐, ๐, and ๐ are the vertices of a triangle and ๐ > 1, show that (a) ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ || ๐โฒ๐โฒ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ and (b) ๐โฒ๐โฒ = ๐๐๐.Use the diagram below when writing your proof.
2. a. Produce a scale drawing of โ๐ฟ๐๐ using either the ratio or parallel method with point ๐ as the center and a
scale factor of 32.
![Page 35: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/35.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 5 GEOMETRY
Lesson 5: Scale Factors Date: 9/26/14 S.35
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b. Use the dilation theorem to predict the length of ๐ฟโฒ๐โฒ, then measure its length directly using a ruler.
c. Does the dilation theorem appear to hold true?
3. Produce a scale drawing of โ๐๐๐ with point ๐ as the center and a scale factor of 14. Use the dilation theorem to
predict ๐โฒ๐โฒ, and then measure its length directly using a ruler. Does the dilation theorem appear to hold true
4. Given the diagram below, determine if โ๐ท๐ธ๐น is a scale drawing of โ๐ท๐บ๐ป. Explain why or why not.
![Page 36: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/36.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 5 GEOMETRY
Lesson 5: Scale Factors Date: 9/26/14 S.36
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Drawing Not to Scale
Problem Set
1. โ๐ด๐ตโฒ๐ถโฒ is a dilation of โ๐ด๐ต๐ถ from vertex ๐ด, and ๐ถ๐ถโฒ = 2. Use the given information in each part and the diagramto find ๐ตโฒ๐ถโฒ.a. ๐ด๐ต = 9, ๐ด๐ถ = 4, and ๐ต๐ถ = 7b. ๐ด๐ต = 4, ๐ด๐ถ = 9, and ๐ต๐ถ = 7c. ๐ด๐ต = 7, ๐ด๐ถ = 9, and ๐ต๐ถ = 4d. ๐ด๐ต = 7, ๐ด๐ถ = 4, and ๐ต๐ถ = 9e. ๐ด๐ต = 4, ๐ด๐ถ = 7, and ๐ต๐ถ = 9f. ๐ด๐ต = 9, ๐ด๐ถ = 7, and ๐ต๐ถ = 4
2. Given the diagram, โ ๐ถ๐ด๐ต โ โ ๐ถ๐น๐ธ. Find ๐ด๐ต.
3. Use the diagram to answer each part below.
a. โ๐๐โฒ๐โฒ is the dilated image of โ๐๐๐ from point ๐ with a scale factor of ๐ > 1. Draw a possible ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
b. โ๐๐โฒโฒ๐โฒโฒ is the dilated image of โ๐๐๐ from point ๐ with a scale factor ๐ > ๐. Draw a possible ๐โฒโฒ๐โฒโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
![Page 37: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/37.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 5 GEOMETRY
Lesson 5: Scale Factors Date: 9/26/14 S.37
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4. Given the diagram to the right, ๐ ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, Area (โ๐ ๐๐) = 15 units2, and Area(โ๐๐๐ ) = 21 units2, find ๐ ๐.
5. Using the information given in the diagram and ๐๐ = 9, find ๐ on ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ such that ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is an altitude. Then find ๐๐ and๐๐.
![Page 38: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/38.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 6 GEOMETRY
Lesson 6: Dilations as Transformations of the Plane Date: 9/26/14 S.38
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Lesson 6: Dilations as Transformations of the Plane
Classwork
Exercises 1โ6
1. Find the center and the angle of the rotation that takes ๐ด๐ต to ๐ดโฒ๐ตโฒ. Find the image ๐โฒ of point ๐ under this rotation.
![Page 39: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/39.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 6 GEOMETRY
Lesson 6: Dilations as Transformations of the Plane Date: 9/26/14 S.39
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2. In the diagram below, โณ ๐ตโฒ๐ถโฒ๐ทโฒ is the image of โณ ๐ต๐ถ๐ท after a rotation about a point ๐ด. What are the coordinates of๐ด, and what is the degree measure of the rotation?
3. Find the line of reflection for the reflection that takes point ๐ด to point ๐ดโฒ. Find the image ๐โฒ under this reflection.
![Page 40: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/40.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 6 GEOMETRY
Lesson 6: Dilations as Transformations of the Plane Date: 9/26/14 S.40
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4. Quinn tells you that the vertices of the image of quadrilateral ๐ถ๐ท๐ธ๐น reflected over the line representing theequation ๐ฆ = โ 3
2 ๐ฅ + 2 are the following: ๐ถโฒ(โ2,3), ๐ทโฒ(0,0), ๐ธโฒ(โ3,โ3), and ๐นโฒ(โ3,3). Do you agree or disagree with Quinn? Explain.
5. A translation takes ๐ด to ๐ดโฒ. Find the image ๐โฒ and pre-image ๐โฒโฒ of point ๐ under this translation. Find a vector thatdescribes the translation.
![Page 41: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/41.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 6 GEOMETRY
Lesson 6: Dilations as Transformations of the Plane Date: 9/26/14 S.41
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6. The point ๐ถโฒ is the image of point ๐ถ under a translation of the plane along a vector.
a. Find the coordinates of ๐ถ if the vector used for the translation is ๐ต๐ด๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ .
b. Find the coordinates of ๐ถ if the vector used for the translation is ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ .
![Page 42: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/42.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 6 GEOMETRY
Lesson 6: Dilations as Transformations of the Plane Date: 9/26/14 S.42
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Exercises 7โ9
7. A dilation with center ๐ and scale factor ๐ takes ๐ด to ๐ดโฒ and ๐ต to ๐ตโฒ. Find the center ๐ and estimate the scale factor๐.
8. Given a center ๐, scale factor ๐, and points ๐ด and ๐ต, find the points ๐ดโฒ = ๐ท๐,๐(๐ด) and ๐ตโฒ = ๐ท๐,๐(๐ต). Compare
length ๐ด๐ต with length ๐ดโฒ๐ตโฒ by division; in other words, find ๐ดโฒ๐ตโฒ
๐ด๐ต. How does this number compare to ๐?
![Page 43: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/43.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 6 GEOMETRY
Lesson 6: Dilations as Transformations of the Plane Date: 9/26/14 S.43
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9. Given a center ๐, scale factor ๐, and points ๐ด, ๐ต, and ๐ถ, find the points ๐ดโฒ = ๐ท๐,๐(๐ด), ๐ตโฒ = ๐ท๐,๐(๐ต), and ๐ถโฒ =๐ท๐,๐(๐ถ). Compare mโ ๐ด๐ต๐ถ with โ ๐ดโฒ๐ตโฒ๐ถโฒ . What do you find?
![Page 44: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/44.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 6 GEOMETRY
Lesson 6: Dilations as Transformations of the Plane Date: 9/26/14 S.44
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Problem Set
1. In the diagram below, ๐ดโฒ is the image of ๐ด under a single transformation of the plane. Use the given diagram toshow your solutions to parts (a)โ(d).
a. Describe the translation that maps ๐ด โ ๐ดโฒ, and then use the translation to locate ๐โฒ, the image of ๐.
b. Describe the reflection that maps ๐ด โ ๐ดโฒ, and then use the reflection to locate ๐โฒ, the image of ๐.
c. Describe a rotation that maps ๐ด โ ๐ดโฒ, and then use your rotation to locate ๐โฒ, the image of ๐.d. Describe a dilation that maps ๐ด โ ๐ดโฒ, and then use your dilation to locate ๐โฒ, the image of ๐.
Lesson Summary
There are two major classes of transformations; those that are distance-preserving (translations,reflections, rotations) and those that are not (dilations).
Like rigid motions, dilations involve a rule assignment for each point in the plane and also have inversefunctions that return each dilated point back to itself.
![Page 45: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/45.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 6 GEOMETRY
Lesson 6: Dilations as Transformations of the Plane Date: 9/26/14 S.45
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2. On the diagram below, ๐ is a center of dilation and ๐ด๐ท๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ is a line not through ๐. Choose two points ๐ต and ๐ถ on ๐ด๐ท๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโbetween ๐ด and ๐ท.
a. Dilate ๐ด, ๐ต, ๐ถ, and ๐ท from ๐ using scale factor ๐ = 12. Label the images ๐ดโฒ, ๐ตโฒ, ๐ถโฒ, and ๐ทโฒ, respectively.
b. Dilate ๐ด, ๐ต, ๐ถ, and ๐ท from ๐ using scale factor ๐ = 2. Label the images ๐ดโฒโฒ, ๐ตโฒโฒ, ๐ถโฒโฒ, and ๐ทโฒโฒ, respectively.
c. Dilate ๐ด, ๐ต, ๐ถ, and ๐ท from ๐ using scale factor ๐ = 3. Label the images ๐ดโฒโฒโฒ, ๐ตโฒโฒโฒ, ๐ถโฒโฒโฒ, and ๐ทโฒโฒโฒ, respectively.
d. Draw a conclusion about the effect of a dilation on a line segment based on the diagram that you drew.Explain.
3. Write the inverse transformation for each of the following so that the composition of the transformation with itsinverse will map a point to itself on the plane.a. ๐๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโb. ๐๐ด๐ต๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโc. ๐ ๐ถ,45
d. ๐ท๐,๐
4. Given ๐(1,3), ๐(โ4,โ4), and ๐(โ3,6) on the coordinate plane, perform a dilation of โณ ๐๐๐ from center ๐(0,0)with a scale factor of 3
2. Determine the coordinates of images of points ๐, ๐, and ๐, and describe how the
coordinates of the image points are related to the coordinates of the pre-image points.
5. Points ๐ต, ๐ถ, ๐ท, ๐ธ, ๐น, and ๐บ are dilated images of ๐ด from center ๐ with scale factors 2, 3, 4, 5, 6, and 7, respectively.Are points ๐, ๐, ๐, ๐, ๐, ๐, and ๐ all dilated images of ๐ under the same respective scale factors? Explain why orwhy not.
![Page 46: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/46.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 6 GEOMETRY
Lesson 6: Dilations as Transformations of the Plane Date: 9/26/14 S.46
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6. Find the center and scale factor that takes ๐ด to ๐ดโฒ and ๐ต to ๐ตโฒ, if a dilation exists.
7. Find the center and scale factor that takes ๐ด to ๐ดโฒ and ๐ต to ๐ตโฒ, if a dilation exists.
![Page 47: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/47.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 7 GEOMETRY
Lesson 7: How Do Dilations Map Segments? Date: 9/26/14 S.47
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Lesson 7: How Do Dilations Map Segments?
Classwork
Opening Exercise
a. Is a dilated segment still a segment? If the segment is transformed under a dilation, explain how.
b. Dilate the segment ๐๐ by a scale factor of 2 from center ๐.
![Page 48: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/48.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 7 GEOMETRY
Lesson 7: How Do Dilations Map Segments? Date: 9/26/14 S.48
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i. Is the dilated segment ๐โฒ๐โฒ a segment?
ii. How, if at all, has the segment ๐๐ been transformed?
Example 1
Case 1. Consider the case where the scale factor of dilation is ๐ = 1. Will a dilation from center ๐ map segment ๐๐ to a segment ๐โฒ๐โฒ? Explain.
Example 2
Case 2. Consider the case where a line ๐๐ contains the center of the dilation. Will a dilation from the center with scale factor ๐ โ 1 map the segment ๐๐ to a segment ๐โฒ๐โฒ? Explain.
![Page 49: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/49.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 7 GEOMETRY
Lesson 7: How Do Dilations Map Segments? Date: 9/26/14 S.49
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Example 3
Case 3. Consider the case where ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ does not contain the center ๐ of the dilation and the scale factor ๐ of the dilation is not equal to 1; then we have the situation where the key points ๐, ๐, and ๐ form โณ ๐๐๐. The scale factor not being equal to 1 means that we must consider scale factors such that 0 < ๐ < 1 and ๐ > 1. However, the proofs for each are similar, so we will focus on the case when 0 < ๐ < 1.
When we dilate points ๐ and ๐ from center ๐ by scale factor 0 < ๐ < 1, as shown, what do we know about points ๐โฒ and ๐โฒ?
We will use the fact that the line segment ๐โฒ๐โฒ splits the sides of โณ ๐๐๐ proportionally and that the lines containing ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐โฒ๐โฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ are parallel to prove that a dilation maps segments to segments. Because we already know what happens when points ๐ and ๐ are dilated, consider another point ๐ that is on the segment ๐๐. After dilating ๐ from center ๐ by scale factor ๐ to get the point ๐ โฒ, does ๐ โฒ lie on the segment ๐โฒ๐โฒ?
Putting together the preliminary dilation theorem for segments with the dilation theorem, we get:
DILATION THEOREM FOR SEGMENTS: A dilation ๐ท๐,๐ maps a line segment ๐๐ to a line segment ๐โฒ๐โฒ sending the endpoints to
the endpoints so that ๐โฒ๐โฒ = ๐๐๐. Whenever the center ๐ does not lie in line ๐๐ and ๐ โ 1, we conclude ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ || ๐โฒ๐โฒ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ . Whenever the center ๐ lies in ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ or if ๐ = 1, we conclude ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ = ๐โฒ๐โฒ๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ .
As an aside, observe that dilation maps parallel line segments to parallel line segments. Further, a dilation maps a directed line segment to a directed line segment that points in the same direction.
![Page 50: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/50.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 7 GEOMETRY
Lesson 7: How Do Dilations Map Segments? Date: 9/26/14 S.50
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Example 4
Now look at the converse of the dilation theorem for segments: If ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ are line segments of different lengths in the plane, then there is a dilation that maps one to the other if and only if ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ = ๐ ๐๏ฟฝโ๏ฟฝ๏ฟฝโ or ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ || ๐ ๐๏ฟฝโ๏ฟฝ๏ฟฝโ .
Based on Examples 2 and 3, we already know that a dilation maps a segment ๐๐ to another line segment, say ๐ ๐,๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ so that ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ = ๐ ๐๏ฟฝโ๏ฟฝ๏ฟฝโ (Example 2) or ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ || ๐ ๐๏ฟฝโ๏ฟฝ๏ฟฝโ (Example 3). If ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ || ๐ ๐๏ฟฝโ๏ฟฝ๏ฟฝโ , then, because ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ are different lengths in the plane, they are bases of a trapezoid, as shown.
Since ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ are segments of different lengths, then the non-base sides of the trapezoid are not parallel, and the lines containing them will meet at a point ๐ as shown.
Recall that we want to show that a dilation will map ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to ๐ ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Explain how to show it.
The case when the segments ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ are such that ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ = ๐ ๐๏ฟฝโ๏ฟฝ๏ฟฝโ is left as an exercise.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 7 GEOMETRY
Lesson 7: How Do Dilations Map Segments? Date: 9/26/14 S.51
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Exercises 1โ2
In the following exercises, you will consider the case where the segment and its dilated image belong to the same line; that is, when ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ are such that ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ = ๐ ๐๏ฟฝโ๏ฟฝ๏ฟฝโ .
1. Consider points ๐, ๐, ๐ , and ๐ on a line, where ๐ = ๐ , as shown below. Show there is a dilation that maps ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to๐ ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Where is the center of the dilation?
2. Consider points ๐, ๐, ๐ , and ๐ on a line as shown below where ๐๐ โ ๐ ๐. Show there is a dilation that maps ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to๐ ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Where is the center of the dilation?
![Page 52: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/52.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 7 GEOMETRY
Lesson 7: How Do Dilations Map Segments? Date: 9/26/14 S.52
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Problem Set
1. Draw the dilation of parallelogram ๐ด๐ต๐ถ๐ท from center ๐ using the scale factor ๐ = 2, and then answer the questionsthat follow.
a. Is the image ๐ดโฒ๐ตโฒ๐ถโฒ๐ทโฒ also a parallelogram? Explain.
b. What do parallel lines seem to map to under a dilation?
2. Given parallelogram ๐ด๐ต๐ถ๐ท with ๐ด(โ8,1), ๐ต(2,โ4), ๐ถ(โ3,โ6), and ๐ท(โ13,โ1), perform a dilation of the planecentered at the origin using the following scale factors.
a. Scale factor 12
b. Scale factor 2c. Are the images of parallel line segments under a dilation also parallel? Use your graphs to support your
answer.
Lesson Summary
When a segment is dilated by a scale factor of ๐ = 1, then the segment and its image would be the samelength.
When the points ๐ and ๐ are on a line containing the center, then the dilated points ๐โฒ and ๐โฒ will also becollinear with the center producing an image of the segment that is a segment.
When the points ๐ and ๐ are not collinear with the center, and the segment is dilated by a scale factor of๐ โ 1, then the point ๐โฒ lies on the ray ๐๐โฒ with ๐๐โฒ = ๐ โ ๐๐ and ๐โฒlies on ray ๐๐ with ๐โฒ = ๐ โ ๐๐.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 7 GEOMETRY
Lesson 7: How Do Dilations Map Segments? Date: 9/26/14 S.53
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3. In Lesson 7, Example 3, we proved that a line segment ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, where ๐, ๐, and ๐ are the vertices of a triangle, maps toa line segment ๐โฒ๐โฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ under a dilation with a scale factor ๐ < 1. Using a similar proof, prove that for ๐ not on ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ , adilation with center ๐ and scale factor ๐ > 1 maps a point ๐ on ๐๐ to a point ๐ โฒ on line ๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ .
4. On the plane, ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ดโฒ๐ตโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ด๐ต โ ๐ดโฒ๐ตโฒ. Describe a dilation mapping ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to ๐ดโฒ๐ตโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. (Hint: There are 2 cases toconsider.)
5. Only one of Figures ๐ด, ๐ต, or ๐ถ below contains a dilation that maps ๐ด to ๐ดโฒ and ๐ต to ๐ตโฒ. Explain for each figure whythe dilation does or does not exist. For each figure, assume that ๐ด๐ต โ ๐ดโฒ๐ตโฒ.a.
b.
c.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 8 GEOMETRY
Lesson 8: How Do Dilations Map Rays, Lines, and Circles? Date: 9/26/14 S.54
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Lesson 8: How Do Dilations Map Rays, Lines, and Circles?
Classwork
Opening Exercise
a. Is a dilated ray still a ray? If the ray is transformed under a dilation, explain how.
b. Dilate the ray ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ by a scale factor of 2 from center O.
i. Is the figure ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ a ray?
![Page 55: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/55.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 8 GEOMETRY
Lesson 8: How Do Dilations Map Rays, Lines, and Circles? Date: 9/26/14 S.55
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ii. How, if at all, has the segment ๐๐ ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ been transformed?
iii. Will a ray always be mapped to a ray? Explain how you know.
Example 1
Will a dilation about center ๐ and scale factor ๐ = 1 map ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ to ๐โฒ๐โฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ ? Explain.
Example 2
The line that contains ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ does not contain point ๐. Will a dilation about center ๐ and scale factor ๐ โ 1 map ๐๐ ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ to ๐โฒ๐โฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ ?
![Page 56: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/56.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 8 GEOMETRY
Lesson 8: How Do Dilations Map Rays, Lines, and Circles? Date: 9/26/14 S.56
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Example 3
The line that contains ๐๐ ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ contains point ๐. Will a dilation about center ๐ and scale factor ๐ map ray ๐๐ to a ray ๐โฒ๐โฒ?
a. Examine the case where the endpoint ๐ of ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ coincides with the center ๐ of the dilation.
b. Examine the case where the endpoint ๐ of ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ is between ๐ and ๐ on the line containing ๐, ๐, and ๐.
c. Examine the remaining case where the center ๐ of the dilation and point ๐ are on the same side of ๐ on theline containing ๐, ๐, and ๐.
Example 5
Will a dilation about a center ๐ and scale factor ๐ map a circle of radius ๐ onto another circle?
a. Examine the case where the center of the dilation coincides with the center of the circle.
b. Examine the case where the center of the dilation is not the center of the circle; we call this the general case.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 8 GEOMETRY
Lesson 8: How Do Dilations Map Rays, Lines, and Circles? Date: 9/26/14 S.57
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Problem Set
1. In Lesson 8, Example 2, you proved that a dilation with a scale factor ๐ > 1 maps a ray ๐๐ to a ray ๐โฒ๐โฒ. Prove theremaining case that a dilation with scale factor 0 < ๐ < 1 maps a ray ๐๐ to a ray ๐โฒ๐โฒ.
Given the dilation ๐ท๐,๐ , with 0 < ๐ < 1 maps ๐ to ๐โฒ and ๐ to ๐โฒ, prove that ๐ท๐,๐ maps ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ to ๐โฒ๐โฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ .
2. In the diagram below, ๐ดโฒ๐ตโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ is the image of ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ under a dilation from point ๐ with an unknown scale factor, ๐ด mapsto ๐ดโฒ and ๐ต maps to ๐ตโฒ. Use direct measurement to determine the scale factor ๐, and then find the center ofdilation ๐.
3. Draw a line ๐ด๐ต and dilate points ๐ด and ๐ต from center ๐ where ๐ is not on ๐ด๐ต๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ . Use your diagram to explain why aline maps to a line under a dilation with scale factor ๐.
4. Let ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ be a line segment, and let ๐ be a line that is the perpendicular bisector of ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. If a dilation with scale factor๐ maps ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to ๐ดโฒ๐ตโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ (sending ๐ด to ๐ดโฒ and ๐ต to ๐ตโฒ) and also maps line ๐ to line ๐โฒ, show that ๐โฒ is the perpendicularbisector of ๐ดโฒ๐ตโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
5. Dilate circle ๐ถ with radius ๐ถ๐ด from center ๐ with a scale factor ๐ = 12.
Lesson Summary
DILATION THEOREM FOR RAYS: A dilation maps a ray to a ray sending the endpoint to the endpoint.
DILATION THEOREM FOR LINES: A dilation maps a line to a line. If the center ๐ of the dilation lies on the lineor if the scale factor ๐ of the dilation is equal to 1, then the dilation maps the line to the same line.Otherwise, the dilation maps the line to a parallel line.
DILATION THEOREM FOR CIRCLES: A dilation maps a circle to a circle, and maps the center to the center.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 8 GEOMETRY
Lesson 8: How Do Dilations Map Rays, Lines, and Circles? Date: 9/26/14 S.58
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6. In the picture below, the larger circle is a dilation of the smaller circle. Find the center of dilation ๐.
![Page 59: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/59.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 9 GEOMETRY
Lesson 9: How Do Dilations Map Angles? Date: 9/26/14 S.59
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Lesson 9: How Do Dilations Map Angles?
Classwork
Exploratory Challenge/Exercises 1โ4
1. How do dilations map triangles?a. Make a conjecture.
b. Verify your conjecture by experimenting with diagrams and directly measuring angles and lengths of segments.
2. How do dilations map rectangles?
a. Make a conjecture.
b. Verify your conjecture by experimenting with diagrams and directly measuring angles and lengths ofsegments.
![Page 60: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/60.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 9 GEOMETRY
Lesson 9: How Do Dilations Map Angles? Date: 9/26/14 S.60
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3. How do dilations map squares?
a. Make a conjecture.
b. Verify your conjecture by experimenting with diagrams and directly measuring angles and lengths of segments.
4. How do dilations map regular polygons?
a. Make a conjecture.
b. Verify your conjecture by experimenting with diagrams and directly measuring angles and lengths ofsegments.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 9 GEOMETRY
Lesson 9: How Do Dilations Map Angles? Date: 9/26/14 S.61
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Exercises 5โ6
5. Recall what you learned about parallel lines cut by a transversal, specifically about the angles that are formed.
6. A dilation from center ๐ by scale factor ๐ maps โ ๐ต๐ด๐ถ to โ ๐ตโฒ๐ดโฒ๐ถโฒ. Show that ๐โ ๐ต๐ด๐ถ = ๐โ ๐ตโฒ๐ดโฒ๐ถโฒ.
Discussion
The dilation theorem for angles is as follows:
THEOREM: A dilation from center ๐ and scale factor ๐ maps an angle to an angle of equal measure.
We have shown this when the angle and its image intersect at a single point, and that point of intersection is not the vertex of the angle.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 9 GEOMETRY
Lesson 9: How Do Dilations Map Angles? Date: 9/26/14 S.62
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Problem Set
1. Shown below is โณ ๐ด๐ต๐ถ and its image โณ ๐ดโฒ๐ตโฒ๐ถโฒ after it has been dilated from center ๐ by scale factor ๐ = 52. Prove
that the dilation maps โณ ๐ด๐ต๐ถ to โณ ๐ดโฒ๐ตโฒ๐ถโฒ so that ๐โ ๐ด = ๐โ ๐ดโฒ, ๐โ ๐ต = ๐โ ๐ตโฒ, and ๐โ ๐ถ = ๐โ ๐ถโฒ.
2. Explain the effect of a dilation with scale factor ๐ on the length of the base and height of a triangle. How is the areaof the dilated image related to the area of the pre-image?
3. Dilate trapezoid ๐ด๐ต๐ท๐ธ from center ๐ using a scale factor of ๐ = 12.
Lesson Summary
Dilations map angles to angles of equal measure. Dilations map polygonal figures to polygonal figures whose angles are equal in measure to the
corresponding angles of the original figure and whose side lengths are equal to the corresponding sidelengths multiplied by the scale factor.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 9 GEOMETRY
Lesson 9: How Do Dilations Map Angles? Date: 9/26/14 S.63
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4. Dilate kite ๐ด๐ต๐ถ๐ท from center ๐ using a scale factor ๐ = 1 12.
5. Dilate hexagon ๐ท๐ธ๐น๐บ๐ป๐ผ from center ๐ using a scale factor of ๐ = 14.
6. Examine the dilations that you constructed in Problems 2โ5, and describe how each image compares to its pre-image under the given dilation. Pay particular attention to the sizes of corresponding angles and the lengths ofcorresponding sides.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 10 GEOMETRY
Lesson 10: Dividing the Kingโs Foot into 12 Equal Pieces Date: 9/26/14 S.64
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Lesson 10: Dividing the Kingโs Foot into 12 Equal Pieces
Classwork
Opening Exercise
Use a compass to mark off equally spaced points ๐ถ, ๐ท, ๐ธ, and ๐น so that ๐ด๐ต, ๐ต๐ถ, ๐ถ๐ท, ๐ท๐ธ, and ๐ธ๐น are equal in length. Describe the steps you took.
Exploratory Challenge 1
Divide segment ๐ด๐ต into three segments of equal lengths.
![Page 65: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/65.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 10 GEOMETRY
Lesson 10: Dividing the Kingโs Foot into 12 Equal Pieces Date: 9/26/14 S.65
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exercise 1
Divide segment ๐ด๐ต into five segments of equal lengths.
Exploratory Challenge 2
Divide segment ๐ด๐ต into four segments of equal length.
Exercise 2
On a piece of poster paper, draw a segment ๐ด๐ต with a measurement of 1 foot. Use the dilation method to divide ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ into twelve equal-length segments, or into 12 inches.
![Page 66: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/66.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 10 GEOMETRY
Lesson 10: Dividing the Kingโs Foot into 12 Equal Pieces Date: 9/26/14 S.66
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Problem Set
1. Pretend you are the king or queen and that the length of your foot is the official measurement for one foot. Draw aline segment on a piece of paper that is the length of your foot. (You may have to remove your shoe.) Use themethod above to find the length of 1 inch in your kingdom.
2. Using a ruler, draw a segment that is 10 cm. This length is referred to as a decimeter. Use the side splitter methodto divide your segment into ten equal-sized pieces. What should be the length of each of those pieces based onyour construction? Check the length of the pieces using a ruler. Are the lengths of the pieces accurate?
3. Repeat Problem 2 using the dilation method. What should be the length of each of those pieces based on yourconstruction? Check the length of the pieces using a ruler. Are the lengths of the pieces accurate?
4. A portion of a ruler that measured whole centimeters is shown below. Determine the location of 5 23
cm on the portion of the ruler shown.
Lesson Summary
SIDE SPLITTER METHOD: If ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is a line segment, construct a ray ๐ด๐ด1 and mark off ๐ equally spaced points using a compass of fixed radius to get points ๐ด = ๐ด0, ๐ด1, ๐ด2, โฏ,๐ด๐. Construct ๐ด๐๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ that is a side of โณ ๐ด๐ต๐ด๐. Through each point ๐ด1, ๐ด2, โฏ, ๐ด๐โ1, construct line segments ๐ด๐ค๐ต๐ค๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ parallel to ๐ด๐๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ that connect two sides of โณ ๐ด๐ด๐๐ต.
DILATION METHOD: Construct a ray ๐๐ parallel to ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ . On the parallel ray, use a compass to mark off ๐ equally spaced points ๐1, ๐2,โฏ, ๐๐ so that ๐๐๐ โ ๐ด๐ต. Lines ๐ด๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝโ and ๐ต๐๐๏ฟฝโ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ intersect at a point ๐. Construct the rays ๐๐๐ค๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ that meet ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ in points ๐ด๐.
![Page 67: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/67.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 10 GEOMETRY
Lesson 10: Dividing the Kingโs Foot into 12 Equal Pieces Date: 9/26/14 S.67
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
5. Merrick has a ruler that measures in inches only. He is measuring the length of a line segment that is between 8 and9 in. Divide the one-inch section of Merrickโs ruler below into eighths to help him measure the length of thesegment.
6. Use the dilation method to create an equally spaced 3 ร 3 grid in the following square.
![Page 68: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/68.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 10 GEOMETRY
Lesson 10: Dividing the Kingโs Foot into 12 Equal Pieces Date: 9/26/14 S.68
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
7. Use the side splitter method to create an equally spaced 3 ร 3 grid in the following square.
![Page 69: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/69.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 11 GEOMETRY
Lesson 11 Dilations from Different Centers Date: 9/26/14 S.69
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 11: Dilations from Different Centers
Classwork
Exploratory Challenge 1
Drawing 2 and Drawing 3 are both scale drawings of Drawing 1.
a. Determine the scale factor and center for each scale drawing. Take measurements as needed.
b. Is there a way to map Drawing 2 onto Drawing 3 or map Drawing 3 onto Drawing 2?
c. Generalize the parameters of this example and its results.
![Page 70: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/70.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 11 GEOMETRY
Lesson 11 Dilations from Different Centers Date: 9/26/14 S.70
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exercise 1
Triangle ๐ด๐ต๐ถ has been dilated with scale factor 12 from centers ๐1 and ๐2. What can you say about line segments ๐ด1๐ด2,
๐ต1๐ต2, ๐ถ1๐ถ2?
Exploratory Challenge 2
If Drawing 2 is a scale drawing of Drawing 1 with scale factor ๐1, and Drawing 3 is a scale drawing of Drawing 2 with scale factor ๐2, what is the relationship between Drawing 3 and Drawing 1?
a. Determine the scale factor and center for each scale drawing. Take measurements as needed.
![Page 71: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/71.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 11 GEOMETRY
Lesson 11 Dilations from Different Centers Date: 9/26/14 S.71
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
b. What is the scale factor going from Drawing 1 to Drawing 3? Take measurements as needed.
c. Compare the centers of dilations of Drawing 1 (to Drawing 2) and of Drawing 2 (to Drawing 3). What do younotice about these centers relative to the center of the composition of dilations ๐3?
d. Generalize the parameters of this example and its results.
Exercise 2
Triangle ๐ด๐ต๐ถ has been dilated with scale factor 23 from center ๐1 to get triangle ๐ดโฒ๐ตโฒ๐ถโฒ, and then triangle ๐ดโฒ๐ตโฒ๐ถโฒ is
dilated from center ๐2 with scale factor 12 to get triangle ๐ดโฒโฒ๐ตโฒโฒ๐ถโฒโฒ. Describe the dilation that maps triangle ๐ด๐ต๐ถ to
triangle ๐ดโฒโฒ๐ตโฒโฒ๐ถโฒโฒ. Find the center and scale factor for that dilation.
![Page 72: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/72.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 11 GEOMETRY
Lesson 11 Dilations from Different Centers Date: 9/26/14 S.72
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Problem Set
1. In Lesson 7, the dilation theorem for line segments said that if two different length line segments in the plane wereparallel to each other, then a dilation exists mapping one segment onto the other. Explain why the line segmentsmust be different lengths for a dilation to exist.
2. Regular hexagon ๐ดโฒ๐ตโฒ๐ถโฒ๐ทโฒ๐ธโฒ๐นโฒ is the image of regular hexagon ๐ด๐ต๐ถ๐ท๐ธ๐น under a dilation from center ๐1, andregular hexagon ๐ดโฒโฒ๐ตโฒโฒ๐ถโฒโฒ๐ทโฒโฒ๐ธโฒโฒ๐นโฒโฒ is the image of regular hexagon ๐ด๐ต๐ถ๐ท๐ธ๐น under a dilation from center ๐2. Points๐ดโฒ, ๐ตโฒ, ๐ถโฒ, ๐ทโฒ, ๐ธโฒ, and ๐นโฒ are also the images of points ๐ดโฒโฒ, ๐ตโฒโฒ, ๐ถโฒโฒ, ๐ทโฒโฒ, ๐ธโฒโฒ, and ๐นโฒโฒ, respectively, under a translationalong vector ๐ทโฒโฒ๐ทโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ . Find a possible regular hexagon ๐ด๐ต๐ถ๐ท๐ธ๐น.
Lesson Summary
In a series of dilations, the scale factor that maps the original figure onto the final image is the product of all the scale factors in the series of dilations.
![Page 73: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/73.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 11 GEOMETRY
Lesson 11 Dilations from Different Centers Date: 9/26/14 S.73
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
3. A dilation with center ๐1 and scale factor 12
maps figure ๐น to figure ๐นโฒ. A dilation with center ๐2 and scale factor 32
maps figure ๐นโฒ to figure ๐นโฒโฒ. Draw figures ๐นโฒ and ๐นโฒโฒ, and then find the center ๐ and scale factor ๐ of the dilationthat takes ๐น to ๐นโฒโฒ.
4. If a figure ๐ is dilated from center ๐1 with a scale factor ๐1 = 34 to yield image ๐โฒ, and figure ๐โฒ is then dilated from
center ๐2 with a scale factor ๐2 = 43 to yield figure ๐โฒโฒ. Explain why ๐ โ ๐โฒโฒ.
5. A dilation with center ๐1 and scale factor 12 maps figure ๐ป to figure ๐ปโฒ. A dilation with center ๐2 and scale factor 2
maps figure ๐ปโฒ to figure ๐ปโฒโฒ. Draw figures ๐ปโฒ and ๐ปโฒโฒ. Find a vector for a translation that maps ๐ป to ๐ปโฒโฒ.
![Page 74: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/74.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 11 GEOMETRY
Lesson 11 Dilations from Different Centers Date: 9/26/14 S.74
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6. Figure ๐ is dilated from ๐1 with a scale factor ๐1 = 2 to yield ๐โฒ. Figure ๐โฒ is then dilated from center ๐2 with ascale factor ๐2 = 1
4 to yield ๐โฒโฒ.
a. Construct the composition of dilations of figure ๐ described above.b. If you were to dilate figure ๐โฒโฒ, what scale factor would be required to yield an image that is congruent to
figure ๐?
c. Locate the center of dilation that maps ๐โฒโฒ to ๐ using the scale factor that you identified in part (b).
![Page 75: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/75.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 11 GEOMETRY
Lesson 11 Dilations from Different Centers Date: 9/26/14 S.75
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
7. Figures ๐น1 and ๐น2 in the diagram below are dilations of ๐น from centers ๐1 and ๐2, respectively.
a. Find ๐น.
b. If ๐น1 โ ๐น2, what must be true of the scale factors ๐1 and ๐2 of each dilation?
c. If ๐น1 โ ๐น2, what must be true of the scale factors ๐1 and ๐2 of each dilation?
8. Use a coordinate plane to complete each part below using ๐(2,3), ๐(6,6), and ๐(6,โ1).
a. Dilate โณ ๐๐๐ from the origin with a scale factor ๐1 = 2. List the coordinate of image points ๐โฒ, ๐โฒ, and ๐โฒ.
b. Dilate โณ ๐๐๐ from (0,6) with a scale factor of ๐2 = 34. List the coordinates of image points ๐โฒโฒ, ๐โฒโฒ, and ๐โฒโฒ.
c. Find the scale factor, ๐3, from โณ ๐โฒ๐โฒ๐โฒ to โณ ๐โฒโฒ๐โฒโฒ๐โฒโฒ.d. Find the coordinates of the center of dilation that maps โณ ๐โฒ๐โฒ๐โฒ to โณ ๐โฒโฒ๐โฒโฒ๐โฒโฒ.
![Page 76: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/76.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 12 GEOMETRY
Lesson 12: What Are Similarity Transformations, and Why Do We Need Them? Date: 9/26/14 S.76
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Lesson 12: What Are Similarity Transformations, and Why Do We
Need Them?
Classwork
Opening Exercise
Observe Figures 1 and 2 and the images of the intermediate figures between Figures 1 and 2. Figures 1 and 2 are called similar.
What observations can you make about Figures 1 and 2?
Definition:
A ___________________ ___________________ (or ___________________) is a composition of a finite number of dilations or basic rigid motions. The scale factor of a similarity transformation is the product of the scale factors of the dilations in the composition; if there are no dilations in the composition, the scale factor is defined to be 1.
Definition:
Two figures in a plane are _______________________ if there exists a similarity transformation taking one figure onto the other figure.
![Page 77: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/77.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 12 GEOMETRY
Lesson 12: What Are Similarity Transformations, and Why Do We Need Them? Date: 9/26/14 S.77
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Definition Characteristics
Examples Non-Examples
Example 1
Figure ๐โฒ is similar to Figure ๐. Describe a transformation that will map Figure ๐ onto Figure ๐โฒ?
similar
![Page 78: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/78.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 12 GEOMETRY
Lesson 12: What Are Similarity Transformations, and Why Do We Need Them? Date: 9/26/14 S.78
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Exercises 1โ3
1. Figure 1 is similar to Figure 2. Which transformations compose the similarity transformation that maps Figure 1onto Figure 2?
2. Figure ๐ is similar to Figure ๐โฒ. Which transformations compose the similarity transformation that maps ๐ onto ๐โฒ?
3. Figure 1 is similar to Figure 2. Which transformations compose the similarity transformation that maps Figure 1onto Figure 2?
![Page 79: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/79.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 12 GEOMETRY
Lesson 12: What Are Similarity Transformations, and Why Do We Need Them? Date: 9/26/14 S.79
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Example 2
Show that no sequence of basic rigid motions and dilations takes the small figure to the large figure. Take measurements as needed.
Exercises 4โ5
4. Is there a sequence of dilations and basic rigid motions that takes the large figure to the small figure? Takemeasurements as needed.
5. What purpose do transformations serve? Compare and contrast the application of rigid motions to the applicationof similarity transformations.
![Page 80: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/80.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 12 GEOMETRY
Lesson 12: What Are Similarity Transformations, and Why Do We Need Them? Date: 9/26/14 S.80
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Problem Set
1. What is the relationship between scale drawings, dilations, and similar figures?
a. How are scale drawings and dilations alike?
b. How can scale drawings and dilations differ?
c. What is the relationship of similar figures to scale drawings and dilations?
2. Given the diagram below, identify a similarity transformation, if one exists, mapping Figure A onto Figure B. If onedoes not exist, explain why.
3. Teddy correctly identified a similarity transformation with at least one dilation that maps Figure ๐ผ onto Figure ๐ผ๐ผ.Megan correctly identified a congruence transformation that maps Figure ๐ผ onto Figure ๐ผ๐ผ. What must be trueabout Teddyโs similarity transformation?
Lesson Summary
Two figures are similar if there exists a similarity transformation that maps one figure onto the other.
A similarity transformation is a composition of a finite number of dilations or rigid motions.
![Page 81: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/81.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 12 GEOMETRY
Lesson 12: What Are Similarity Transformations, and Why Do We Need Them? Date: 9/26/14 S.81
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
4. Given the coordinate plane shown, identify a similarity transformation, if one exists, mapping ๐ onto ๐. If one doesnot exist, explain why.
5. Given the diagram below, identify a similarity transformation, if one exists, that maps ๐บ onto ๐ป. If one does notexist, explain why. Provide any necessary measurements to justify your answer.
![Page 82: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/82.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 12 GEOMETRY
Lesson 12: What Are Similarity Transformations, and Why Do We Need Them? Date: 9/26/14 S.82
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6. Given the coordinate plane shown, identify a similarity transformation, if one exists, that maps ๐ด๐ต๐ถ๐ท onto๐ดโฒโฒโฒ๐ตโฒโฒโฒ๐ถโฒโฒโฒ๐ทโฒโฒโฒ. If one does not exist, explain why.
7. The diagram below shows a dilation of the planeโฆor does it? Explain your answer.
![Page 83: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/83.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 13 GEOMETRY
Lesson 13: Properties of Similarity Transformations Date: 9/26/14 S.83
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Lesson 13: Properties of Similarity Transformations
Classwork
Example 1
Similarity transformation ๐บ consists of a rotation about the point ๐ by 90ยฐ, followed by a dilation centered at ๐ with scale factor ๐ = 2, and then a reflection across line โ. Find the image of the triangle.
โ
![Page 84: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/84.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 13 GEOMETRY
Lesson 13: Properties of Similarity Transformations Date: 9/26/14 S.84
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Example 2
A similarity transformation ๐บ applied to trapezoid ๐ด๐ต๐ถ๐ท consists of a translation by vector ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ , followed by a reflection across line ๐๐, and then followed by a dilation centered at ๐ with scale factor ๐ = 2. Recall that we can describe the
same sequence using the following notation: ๐ท๐,2 ๏ฟฝ๐๐๐๏ฟฝ๐๐๐(๐ด๐ต๐ถ๐ท)๏ฟฝ๏ฟฝ. Find the image of ๐ด๐ต๐ถ๐ท.
๐๐
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 13 GEOMETRY
Lesson 13: Properties of Similarity Transformations Date: 9/26/14 S.85
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exercise 1
A similarity transformation for triangle ๐ท๐ธ๐น is described by ๐๐๐ ๏ฟฝ๐ท๐ด,12๏ฟฝ๐ ๐ด,90ยฐ(๐ท๐ธ๐น)๏ฟฝ๏ฟฝ. Locate and label the image of
triangle ๐ท๐ธ๐น under the similarity.
๐๐
๐๐
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 13 GEOMETRY
Lesson 13: Properties of Similarity Transformations Date: 9/26/14 S.86
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Problem Set
1. A similarity transformation consists of a reflection over line โ, followed by a dilation from ๐ with a scale factor of๐ = 3
4. Use construction tools to find โณ ๐บโฒโฒ๐ปโฒโฒ๐ผโฒโฒ.
Lesson Summary
Properties of similarity transformations:
1. Distinct points are mapped to distinct points.
2. Each point ๐โฒ in the plane has a pre-image.
3. There is a scale factor ๐ for ๐บ, so that for any pair of points ๐ and ๐ with images ๐โฒ = ๐บ(๐) and๐โฒ = ๐บ(๐), then ๐โฒ๐โฒ = ๐๐๐.
4. A similarity transformation sends lines to lines, rays to rays, line segments to line segments, and parallel lines toparallel lines.
5. A similarity transformation sends angles to angles of equal measure.
6. A similarity transformation maps a circle of radius ๐ to a circle of radius ๐๐ , where ๐ is the scaling factor of thesimilarity transformation.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 13 GEOMETRY
Lesson 13: Properties of Similarity Transformations Date: 9/26/14 S.87
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
2. A similarity transformation consists of a dilation from point ๐ with a scale factor of ๐ = 2 12, followed by a rotation
about ๐ of โ90ยฐ. Use construction tools to find kite ๐ดโฒโฒ๐ตโฒโฒ๐ถโฒโฒ๐ทโฒโฒ.
3. For the Figure ๐, find the image of ๐โ(๐ ๐,90ห ๏ฟฝ๐ท๐,12(๐)๏ฟฝ.
![Page 88: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/88.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 13 GEOMETRY
Lesson 13: Properties of Similarity Transformations Date: 9/26/14 S.88
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
4. A similarity transformation consists of a translation along vector ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ , followed by a rotation of 60ยฐ about ๐, then adilation from ๐ with scale factor ๐ = 1
3. Use construction tools to find โณ ๐โฒโฒโฒ๐โฒโฒโฒ๐โฒโฒโฒ.
5. Given the quarter-circular figure determined by points ๐ด, ๐ต, and ๐ถ, a similarity transformation consists of a โ65ยฐrotation about point ๐ต, followed by a dilation from point ๐ with a scale factor of ๐ = 1
2. Find the image of the figure
determined by points ๐ดโฒโฒ, ๐ตโฒโฒ, and ๐ถโฒโฒ.
Describe a different similarity transformation that would map quarter-circle ๐ด๐ต๐ถ to quarter-circle ๐ดโฒโฒ๐ตโฒโฒ๐ถโฒโฒ.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 13 GEOMETRY
Lesson 13: Properties of Similarity Transformations Date: 9/26/14 S.89
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
6. A similarity transformation consists of a dilation from center ๐ with a scale factor of 12, followed by a rotation of 60ยฐ
about point ๐. Complete the similarity transformation on Figure ๐ to complete the drawing of Figure ๐โฒโฒ.
7. Given Figure ๐ on the coordinate plane shown below, a similarity transformation consists of a dilation from (0,6)with a scale factor of 1
4, followed by a reflection over line ๐ฅ = โ1, then by a vertical translation of 5 units down.
Find the image of Figure ๐ .
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 13 GEOMETRY
Lesson 13: Properties of Similarity Transformations Date: 9/26/14 S.90
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
8. Given โณ ๐ด๐ต๐ถ, with vertices ๐ด(2,โ7), ๐ต(โ2,โ1), and ๐ถ(3,โ4), locate and label the image of the triangle under the
similarity transformation ๐ท๐ตโฒ,12๏ฟฝ๐ ๐ด,120ยฐ๏ฟฝ๐๐ฅ=2(๐ด๐ต๐ถ)๏ฟฝ๏ฟฝ.
9. In Problem 8, describe the relationship of ๐ดโฒโฒโฒ to ๐ด๐ตโฒ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, and explain your reasoning.
10. Given ๐(โ8,3) and quadrilateral ๐ต๐ถ๐ท๐ธ, with ๐ต(โ5,1), ๐ถ(โ6,โ1), ๐ท(โ4,โ1), and ๐ธ(โ4,2), what are thecoordinates of the vertices of the image of ๐ต๐ถ๐ท๐ธ under the similarity transformation ๐๐ฅโ๐๐ฅ๐๐ ๏ฟฝ๐ท๐,3(๐ต๐ถ๐ท๐ธ)๏ฟฝ?
11. Given triangle ๐ด๐ต๐ถ as shown on the diagram of the coordinate plane:
a. Perform a translation so that vertex ๐ด maps to the origin.
b. Next, dilate the image ๐ดโฒ๐ตโฒ๐ถโฒ from the origin using a scale factor of 13.
c. Finally, translate the image ๐ดโฒโฒ๐ตโฒโฒ๐ถโฒโฒ so that the vertex ๐ดโฒโฒ maps to the original point ๐ด.d. Using transformations, describe how the resulting image ๐ดโฒโฒโฒ๐ตโฒโฒ๐ถโฒโฒ relates to the original figure ๐ด๐ต๐ถ.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 13 GEOMETRY
Lesson 13: Properties of Similarity Transformations Date: 9/26/14 S.91
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
12. a. In the coordinate plane, name the single transformation that is the result of the composition of the two
dilations shown below:๐ท(0,0),2 followed by ๐ท(0,4),12
(Hint: Try it!)
b. In the coordinate plane, name the single transformation that is the result of the composition of the twodilations shown below:
๐ท(0,0),2 followed by ๐ท(4,4),12
(Hint: Try it!)
c. Using the results from parts (a) and (b), describe what happens to the origin under both similaritytransformations.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 14 GEOMETRY
Lesson 14: Similarity Date: 9/26/14 S.92
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Lesson 14: Similarity
Classwork
Example 1
We said that for a figure ๐ด in the plane, it must be true that ๐ด~๐ด. Describe why this must be true.
Example 2
We said that for figures ๐ด and ๐ต in the plane so that ๐ด~๐ต, then it must be true that ๐ต~๐ด. Describe why this must be true.
Example 3
Based on the definition of similar, how would you show that any two circles are similar?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 14 GEOMETRY
Lesson 14: Similarity Date: 9/26/14 S.93
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Example 4
Suppose โณ ๐ด๐ต๐ถ โ โณ ๐ท๐ธ๐น and under this correspondence, corresponding angles are equal and corresponding sides are proportional. Does this guarantee that โณ ๐ด๐ต๐ถ and โณ ๐ท๐ธ๐น are similar?
Example 5
a. In the diagram below, โณ ๐ด๐ต๐ถ~ โณ ๐ดโฒ๐ตโฒ๐ถโฒ. Describe a similarity transformation that mapsโณ ๐ด๐ต๐ถ to โณ ๐ดโฒ๐ตโฒ๐ถโฒ.
b. Joel says the sequence must require a dilation and three rigid motions, but Sharon is sure there is a similaritycomposed of just a dilation and just two rigid motions. Who is right?
![Page 94: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/94.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 14 GEOMETRY
Lesson 14: Similarity Date: 9/26/14 S.94
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Problem Set
1. If you are given any two congruent triangles, describe a sequence of basic rigid motions that will take one to theother.
2. If you are given two similar triangles that are not congruent triangles, describe a sequence of dilations and basicrigid motions that will take one to the other.
3. Given two line segments, ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ถ๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, of different lengths, answer the following questions.
a. It is always possible to find a similarity transformation that maps ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to ๐ถ๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ sending ๐ด to ๐ถ and ๐ต to ๐ท.Describe one such similarity transformation.
b. If you are given that ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ถ๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ are not parallel, are not congruent, do not share any points, and do not lie inthe same line, what is the least number of transformations needed in a sequence to map ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to ๐ถ๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ? Whichtransformations make this work?
c. If you performed a similarity transformation that instead takes ๐ด to ๐ท and ๐ต to ๐ถ, either describe whatmistake was made in the similarity transformation, or describe what additional transformation is needed to fixthe error so that ๐ด maps to ๐ถ and ๐ต maps to ๐ท.
4. We claim that similarity is transitive, i.e., that if ๐ด, ๐ต, and ๐ถ are figures in the plane such that ๐ด~๐ต and ๐ต~๐ถ, then๐ด~๐ถ. Describe why this must be true.
5. Given two line segments, ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ถ๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, of different lengths, we have seen that it is always possible to find a similaritytransformation that maps ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to ๐ถ๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, sending ๐ด to ๐ถ and ๐ต to ๐ท with one rotation and one dilation. Can you do thiswith one reflection and one dilation?
6. Given two triangles, โณ ๐ด๐ต๐ถ~ โณ ๐ท๐ธ๐น, is it always possible to rotate โณ ๐ด๐ต๐ถ so that the sides of โณ ๐ด๐ต๐ถ are parallel tothe corresponding sides in โณ ๐ท๐ธ๐น; i.e., ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ท๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, etc.?
Lesson Summary
Similarity is reflexive because a figure is similar to itself.
Similarity is symmetric because once a similarity transformation is determined to take a figure to another, there are inverse transformations that will take the figure back to the original.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 15 GEOMETRY
Lesson 15: The Angle-Angle (AA) Criterion for Two Triangles to be Similar Date: 9/26/14 S.95
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Lesson 15: The Angle-Angle (AA) Criterion for Two Triangles to be
Similar
Classwork
Exercises 1โ5
1. Draw two triangles of different sizes with two pairs of equal angles. Then measure the lengths of the correspondingsides to verify that the ratio of their lengths is proportional. Use a ruler, compass, or protractor, as necessary.
2. Are the triangles you drew in Exercise 1 similar? Explain.
3. Why is it that you only needed to construct triangles where two pairs of angles were equal and not three?
4. Why were the ratios of the corresponding sides proportional?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 15 GEOMETRY
Lesson 15: The Angle-Angle (AA) Criterion for Two Triangles to be Similar Date: 9/26/14 S.96
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5. Do you think that what you observed will be true when you construct a pair of triangles with two pairs of equalangles? Explain.
Exercises 6โ10
6. Draw another two triangles of different sizes with two pairs of equal angles. Then measure the lengths of thecorresponding sides to verify that the ratio of their lengths is proportional. Use a ruler, compass, or protractor, asnecessary.
7. Are the triangles shown below similar? Explain. If the triangles are similar, identify any missing angle and sidelength measures.
28
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 15 GEOMETRY
Lesson 15: The Angle-Angle (AA) Criterion for Two Triangles to be Similar Date: 9/26/14 S.97
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8. Are the triangles shown below similar? Explain. If the triangles are similar identify any missing angle and side lengthmeasures.
9. The triangles shown below are similar. Use what you know about similar triangles to find the missing side lengths ๐ฅand ๐ฆ.
![Page 98: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/98.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 15 GEOMETRY
Lesson 15: The Angle-Angle (AA) Criterion for Two Triangles to be Similar Date: 9/26/14 S.98
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10. The triangles shown below are similar. Write an explanation to a student, Claudia, of how to find the lengths of ๐ฅand ๐ฆ.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 15 GEOMETRY
Lesson 15: The Angle-Angle (AA) Criterion for Two Triangles to be Similar Date: 9/26/14 S.99
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Problem Set
1. In the figure to the right, โณ ๐ฟ๐๐~ โณ๐๐๐ฟ.
a. Classify โณ ๐ฟ๐๐ based on what you know about similar triangles and justify your reasoning.
b. If ๐โ ๐ = 20ยฐ, find the remaining angles in the diagram.
2. In the diagram below, โณ ๐ด๐ต๐ถ~ โณ ๐ด๐น๐ท. Determine whether the following statements must be true from the giveninformation, and explain why.a. โณ ๐ถ๐ด๐ต~ โณ ๐ท๐ด๐นb. โณ ๐ด๐ท๐น~ โณ ๐ถ๐ด๐ตc. โณ ๐ต๐ถ๐ด~ โณ ๐ด๐ท๐นd. โณ ๐ด๐ท๐น~ โณ ๐ด๐ถ๐ต
3. In the diagram below, ๐ท is the midpoint of ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, ๐น is the midpoint of ๐ต๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, and ๐ธ is the midpoint of ๐ด๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Prove thatโณ ๐ด๐ต๐ถ ~ โณ ๐น๐ธ๐ท.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 15 GEOMETRY
Lesson 15: The Angle-Angle (AA) Criterion for Two Triangles to be Similar Date: 9/26/14 S.100
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4. Use the diagram below to answer each part.
a. If ๐ด๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ธ๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ธ๐น๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, and ๐ถ๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ท๐น๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, prove that the triangles are similar.b. The triangles are not congruent. Find the dilation that takes one to the other.
5. Given trapezoid ๐ด๐ต๐ท๐ธ, and ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ธ๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, prove that โณ ๐ด๐น๐ต ~ โณ ๐ท๐ธ๐น.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 GEOMETRY
Lesson 16: Between-Figure and Within-Figure Ratios Date: 9/26/14 S.101
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Lesson 16: Between-Figure and Within-Figure Ratios
Classwork
Opening Exercise
At a certain time of day, a 12 meter flagpole casts an 8 m shadow. Write an equation that would allow you to find the height, โ, of the tree that uses the length, ๐ , of the treeโs shadow.
Example 1
Given ๐ฅ๐ด๐ต๐ถ ~ ๐ฅ๐ดโฒ๐ตโฒ๐ถโฒ, find the missing side lengths.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 GEOMETRY
Lesson 16: Between-Figure and Within-Figure Ratios Date: 9/26/14 S.102
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Example 2
In the diagram above a large flagpole stands outside of an office building. Marquis realizes that when he looks up from the ground, 60 m away from the flagpole, that the top of the flagpole and the top of the building line up. If the flagpole is 35 m tall, and Marquis is 170 m from the building, how tall is the building?
a. Are the triangles in the diagram similar? Explain.
b. Determine the height of the building using what you know about scale factors.
c. Determine the height of the building using ratios between similar figures.
d. Determine the height of the building using ratios within similar figures.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 GEOMETRY
Lesson 16: Between-Figure and Within-Figure Ratios Date: 9/26/14 S.103
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Example 3
The following right triangles are similar. We will determine the unknown side lengths by using ratios within the first triangle. For each of the triangles below we define the base as the horizontal length of the triangle and the height as the vertical length.
a. Write and find the value of the ratio that compares the height to the hypotenuse of the leftmost triangle.
b. Write and find the value of the ratio that compares the base to the hypotenuse of the leftmost triangle.
c. Write and find the value of the ratio that compares the height to the base of the leftmost triangle.
d. Use the triangle with lengths 3โ 4โ 5 and triangle ๐ด to answer the following questions.
i. Which ratio can be used to determine the height of triangle ๐ด?
ii. Which ratio can be used to determine the hypotenuse of triangle ๐ด?
iii. Find the unknown lengths of triangle ๐ด.
![Page 104: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/104.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 GEOMETRY
Lesson 16: Between-Figure and Within-Figure Ratios Date: 9/26/14 S.104
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e. Use the triangle with lengths 3โ 4โ 5 and triangle ๐ต to answer the following questions.
i. Which ratio can be used to determine the base of triangle ๐ต?
ii. Which ratio can be used to determine the hypotenuse of triangle ๐ต?
iii. Find the unknown lengths of Triangle ๐ต.
f. Use the triangle with lengths 3โ 4โ 5 and triangle ๐ถ to answer the following questions.
i. Which ratio can be used to determine the height of triangle ๐ถ?
ii. Which ratio can be used to determine the base of triangle ๐ถ?
iii. Find the unknown lengths of triangle ๐ถ.
g. Explain the relationship of the ratio of the corresponding sides within a figure to the ratio of correspondingsides within a similar figure.
h. How does the relationship you noted in part (g) allow you to determine the length of an unknown side of atriangle?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 GEOMETRY
Lesson 16: Between-Figure and Within-Figure Ratios Date: 9/26/14 S.105
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Problem Set
1. โ๐ท๐ธ๐น ~ โ๐ด๐ต๐ถ. All side length measurements are in centimeters. Use between ratios and/or within ratios todetermine the unknown side lengths.
2. Given โ๐ด๐ต๐ถ ~ โ๐๐๐, answer the following questions.
a. Write and find the value of the ratio that comparesthe height ๐ด๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to the hypotenuse of โ๐ด๐ต๐ถ.
b. Write and find the value of the ratio that comparesthe base ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to the hypotenuse of โ๐ด๐ต๐ถ.
c. Write and find the value of the ratio that comparesthe height ๐ด๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to the base ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ of โ๐ด๐ต๐ถ.
d. Use within-figure ratios to find the correspondingheight of โ๐๐๐.
e. Use within-figure ratios to find the hypotenuse ofโ๐๐๐.
3. Right triangles ๐ด, ๐ต, ๐ถ, and ๐ท are similar. Determine the unknown side lengths of each triangle by using ratios ofside lengths within triangle ๐ด.
a. Write and find the value of the ratio that compares the height to the hypotenuse of triangle ๐ด.
b. Write and find the value of the ratio that compares the base to the hypotenuse of triangle ๐ด.
c. Write and find the value of the ratio that compares the height to the base of triangle ๐ด.
d. Which ratio can be used to determine the height of triangle ๐ต? Find the height of triangle ๐ต.e. Which ratio can be used to determine the base of triangle ๐ต? Find the base of triangle ๐ต.
f. Find the unknown lengths of triangle ๐ถ.
g. Find the unknown lengths of triangle ๐ท.
A B
C
D
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 GEOMETRY
Lesson 16: Between-Figure and Within-Figure Ratios Date: 9/26/14 S.106
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h. Triangle ๐ธ is also similar to triangles ๐ด, ๐ต, ๐ถ, and ๐ท. Find the lengths of the missing sides in terms of ๐ฅ.
4. Brian is photographing the Washington Monument and wonders how tall the monument is. Brian places his5 ft. camera tripod approximately 100 yd. from the base of the monument. Lying on the ground, he visually alignsthe top of his tripod with the top of the monument and marks his location on the ground approximately 2 ft. 9 in.from the center of his tripod. Use Brianโs measurements to approximate the height of the Washington Monument.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 16 GEOMETRY
Lesson 16: Between-Figure and Within-Figure Ratios Date: 9/26/14 S.107
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5. Catarinaโs boat has come untied and floated away on the lake. She is standing atop a cliff that is 35 feet above thewater in a lake. If she stands 10 feet from the edge of the cliff, she can visually align the top of the cliff with thewater at the back of her boat. He eye level is 5 1
2 feet above the ground. Approximately how far out from the cliff is
Catarinaโs boat?
6. Given the diagram below and โ๐ด๐ต๐ถ~โ๐๐๐, find the unknown lengths ๐ฅ, 2๐ฅ, and 3๐ฅ.
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Lesson 17 The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar
Date: 9/26/14 S.108
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 17 GEOMETRY
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Lesson 17: The Side-Angle-Side (SAS) and Side-Side-Side (SSS)
Criteria for Two Triangles to be Similar
Classwork
Opening Exercise
a. Choose three lengths that represent the sides of a triangle. Draw the triangle with your chosen lengths usingconstruction tools.
b. Multiply each length in your original triangle by 2 to get three corresponding lengths of sides for a secondtriangle. Draw your second triangle using construction tools.
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Lesson 17 The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar
Date: 9/26/14 S.109
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 17 GEOMETRY
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c. Do your constructed triangles appear to be similar? Explain your answer.
d. Do you think that the triangles can be shown similar without knowing the angle measures?
Exploratory Challenge 1/Exercises 1โ2
1. Examine the figure and answer the questions to determine whether or not the triangles shown are similar.
a. What information is given about the triangles in Figure 1?
b. How can the information provided be used to determine whether โณ ๐ด๐ต๐ถ is similar to โณ ๐ด๐ตโฒ๐ถโฒ?
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Lesson 17 The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar
Date: 9/26/14 S.110
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 17 GEOMETRY
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c. Compare the corresponding side lengths of โณ ๐ด๐ต๐ถ and โณ ๐ด๐ตโฒ๐ถโฒ. What do you notice?
d. Based on your work in parts (a)โ(c), draw a conclusion about the relationship between โณ ๐ด๐ต๐ถ and โณ ๐ด๐ตโฒ๐ถโฒ.Explain your reasoning.
2. Examine the figure, and answer the questions to determine whether or not the triangles shown are similar.
a. What information is given about the triangles in Figure 2?
![Page 111: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/111.jpg)
Lesson 17 The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar
Date: 9/26/14 S.111
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 17 GEOMETRY
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b. How can the information provided be used to determine whether โณ ๐๐๐ is similar to โณ ๐๐โฒ๐ โฒ?
c. Compare the corresponding side lengths of โณ ๐๐๐ and โณ ๐๐โฒ๐ โฒ. What do you notice?
d. Based on your work in parts (a)โ(c), draw a conclusion about the relationship between โณ ๐๐๐ and โณ ๐๐โฒ๐ โฒ.Explain your reasoning.
Exploratory Challenge 2/Exercises 3โ4
3. Examine the figure and answer the questions to determine whether or not the triangles shown are similar.
a. What information is given about the triangles in Figure 3?
![Page 112: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/112.jpg)
Lesson 17 The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar
Date: 9/26/14 S.112
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 17 GEOMETRY
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b. How can the information provided be used to determine whether โณ ๐ด๐ต๐ถ is similar to โณ ๐ด๐ตโฒ๐ถโฒ?
c. Compare the corresponding side lengths of โณ ๐ด๐ต๐ถ and โณ ๐ด๐ตโฒ๐ถโฒ. What do you notice?
d. Based on your work in parts (a)โ(c), make a conjecture about the relationship between โณ ๐ด๐ต๐ถ and โณ ๐ด๐ตโฒ๐ถโฒ.Explain your reasoning.
4. Examine the figure and answer the questions to determine whether or not the triangles shown are similar.
![Page 113: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/113.jpg)
Lesson 17 The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar
Date: 9/26/14 S.113
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 17 GEOMETRY
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a. What information is given about the triangles in Figure 4?
b. How can the information provided be used to determine whether โณ ๐ด๐ต๐ถ is similar to โณ ๐ด๐ตโฒ๐ถโฒ?
c. Compare the corresponding side lengths of โณ ๐ด๐ต๐ถ and โณ ๐ด๐ตโฒ๐ถโฒ. What do you notice?
d. Based on your work in parts (a)โ(c), make a conjecture about the relationship between โณ ๐ด๐ต๐ถ and โณ ๐ด๐ตโฒ๐ถโฒ.Explain your reasoning.
Exercises 5โ10
5. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement.
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Lesson 17 The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar
Date: 9/26/14 S.114
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 17 GEOMETRY
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6. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement.
7. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement.
8. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement.
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Lesson 17 The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar
Date: 9/26/14 S.115
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 17 GEOMETRY
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9. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement.
10. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement.
![Page 116: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/116.jpg)
Lesson 17 The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar
Date: 9/26/14 S.116
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 17 GEOMETRY
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Problem Set
1. For each part (a) through (d) below, state which of the three triangles, if any, are similar and why.
a.
b.
c.
d.
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Lesson 17 The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar
Date: 9/26/14 S.117
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 17 GEOMETRY
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2. For each given pair of triangles, determine if the triangles are similar or not, and provide your reasoning. If thetriangles are similar, write a similarity statement relating the triangles.
a.
b.
c.
d.
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Lesson 17 The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar
Date: 9/26/14 S.118
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 17 GEOMETRY
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3. For each pair of similar triangles below, determine the unknown lengths of the sides labeled with letters.
a.
b.
4. Given that ๐ด๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ต๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ intersect at ๐ธ, and ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ถ๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, show that โ๐ด๐ต๐ธ~โ๐ท๐ถ๐ธ.
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Lesson 17 The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar
Date: 9/26/14 S.119
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 17 GEOMETRY
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5. Given ๐ต๐ธ = 11, ๐ธ๐ด = 11, ๐ต๐ท = 7, and ๐ท๐ถ = 7, show that โ๐ต๐ธ๐ท~โ๐ต๐ด๐ถ.
.
6. Given the diagram below, ๐ is on ๐ ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ is on ๐ ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, ๐๐ = 2, ๐๐ = 6, ๐๐ = 9, and ๐๐ = 4.
a. Show that โ๐ ๐๐~โ๐ ๐๐.
b. Find ๐ ๐ and ๐ ๐.
7. One triangle has a 120ยฐ angle, and a second triangle has a 65ยฐ angle. Is it possible that the two triangles are similar?Explain why or why not.
8. A right triangle has a leg that is 12 cm long, and another right triangle has a leg that is 6 cm long. Are the twotriangles similar or not? If so, explain why. If not, what other information would be needed to show they aresimilar?
9. Given the diagram below, ๐ฝ๐ป = 7.5, ๐ป๐พ = 6, and ๐พ๐ฟ = 9, is there a pair of similar triangles? If so, write asimilarity statement and explain why. If not, explain your reasoning.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 18 GEOMETRY
Lesson 18: Similarity and the Angle Bisector Theorem Date: 9/26/14 S.120
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Lesson 18: Similarity and the Angle Bisector Theorem
Classwork
Opening Exercise
a. What is an angle bisector?
b. Describe the angle relationships formed when parallel lines are cut by a transversal.
c. What are the properties of an isosceles triangle?
Discussion
In the diagram below, the angle bisector of โ ๐ด in โณ ๐ด๐ต๐ถ meets side ๐ต๐ถ at point ๐ท. Does the angle bisector create any observable relationships with respect to the side lengths of the triangle?
![Page 121: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/121.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 18 GEOMETRY
Lesson 18: Similarity and the Angle Bisector Theorem Date: 9/26/14 S.121
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Exercises 1โ4
1. The sides of a triangle are 8, 12, and 15. An angle bisector meets the side of length 15. Find the lengths ๐ฅ and ๐ฆ.Explain how you arrived at your answers.
2. The sides of a triangle are 8, 12, and 15. An angle bisector meets the side of length 12. Find the lengths ๐ฅ and ๐ฆ.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 18 GEOMETRY
Lesson 18: Similarity and the Angle Bisector Theorem Date: 9/26/14 S.122
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3. The sides of a triangle are 8, 12, and 15. An angle bisector meets the side of length 8. Find the lengths ๐ฅ and ๐ฆ.
4. The angle bisector of an angle splits the opposite side of a triangle into lengths 5 and 6. The perimeter of thetriangle is 33. Find the lengths of the other two sides.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 18 GEOMETRY
Lesson 18: Similarity and the Angle Bisector Theorem Date: 9/26/14 S.123
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Problem Set
1. The sides of a triangle have lengths of 5, 8, and 6 12. An angle bisector meets the side of length 6 1
2. Find the lengths
๐ฅ and ๐ฆ.
2. The sides of a triangle are 10 12, 16 1
2, and 9. An angle bisector meets the side of length 9. Find the lengths ๐ฅ and ๐ฆ.
3. In the diagram of triangle ๐ท๐ธ๐น below, ๐ท๐บ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is an angle bisector, ๐ธ = 8, ๐ท๐น = 6, and ๐ธ๐น = 8 16. Find ๐น๐บ and ๐ธ๐บ.
4. โ ๐ต๐ด๐ท โ โ ๐ท๐ด๐ถ, show that ๐ต๐ท:๐ต๐ด = ๐ถ๐ท:๐ถ๐ด.
5. The perimeter of triangle ๐ฟ๐๐ is 32 cm. ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is the angle bisector of angle ๐, ๐ฟ๐ = 3 cm, and ๐๐ = 5 cm. Find ๐ฟ๐and ๐๐.
6. Given ๐ถ๐ท = 3, ๐ท๐ต = 4, ๐ต๐น = 4, ๐น๐ธ = 5, ๐ด๐ต = 6, and โ ๐ถ๐ด๐ท โ โ ๐ท๐ด๐ต โ โ ๐ต๐ด๐น โ โ ๐น๐ด๐ธ, find the perimeter ofquadrilateral ๐ด๐ธ๐ต๐ถ
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 18 GEOMETRY
Lesson 18: Similarity and the Angle Bisector Theorem Date: 9/26/14 S.124
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7. If ๐ด๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ meets ๐ต๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ at ๐ท such that ๐ถ๐ท:๐ต๐ท = ๐ถ๐ด:๐ต๐ด, show that โ ๐ถ๐ด๐ท โ โ ๐ต๐ด๐ท. Explain how this proof relates to theangle bisector theorem.
8. In the diagram below, ๐ธ๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โ ๐ท๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, ๐ต๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ bisects โ ๐ด๐ต๐ถ, ๐ด๐ท = 4, and ๐ท๐ถ = 8. Prove that โ๐ด๐ท๐ต~โ๐ถ๐ธ๐ต.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 19 GEOMETRY
Lesson 19: Families of Parallel Lines and Circumference of the Earth Date: 9/26/14 S.125
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Lesson 19: Families of Parallel Lines and the Circumference of the
Earth
Classwork
Opening Exercise
Show ๐ฅ: ๐ฆ = ๐ฅโฒ: ๐ฆโฒ is equivalent to ๐ฅ:๐ฅโฒ = ๐ฆ: ๐ฆโฒ.
Exercises 1โ2
Lines that appear to be parallel are in fact parallel.
1.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 19 GEOMETRY
Lesson 19: Families of Parallel Lines and Circumference of the Earth Date: 9/26/14 S.126
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2.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 19 GEOMETRY
Lesson 19: Families of Parallel Lines and Circumference of the Earth Date: 9/26/14 S.127
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Problem Set
1. Given the diagram shown, ๐ด๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐บ๐ฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ฟ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, and ๐ด๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ต๐ ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ถ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ โฅ ๐ท๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Use the additional information given ineach part below to answer the questions:
a. If ๐บ๐ฟ = 4, what is ๐ป๐?
b. If ๐บ๐ฟ = 4, ๐ฟ๐ = 9, and ๐๐ = 5, what is ๐๐?
c. Using information from part (b), if ๐ถ๐ผ = 18, what is ๐๐?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 19 GEOMETRY
Lesson 19: Families of Parallel Lines and Circumference of the Earth Date: 9/26/14 S.128
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2. Use your knowledge about families of parallel lines to find the coordinates of point ๐ on the coordinate planebelow.
3. ๐ด๐ถ๐ท๐ต and ๐น๐ถ๐ท๐ธ are both trapezoids with bases ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, ๐น๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, and ๐ถ๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. The perimeter of trapezoid ๐ด๐ถ๐ท๐ต is 24 12. If the
ratio of ๐ด๐น:๐น๐ถ is 1: 3, and ๐ธ๐ท = 5 58, find ๐ด๐น, ๐น๐ถ, and ๐ต๐ธ.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 19 GEOMETRY
Lesson 19: Families of Parallel Lines and Circumference of the Earth Date: 9/26/14 S.129
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4. Given the diagram and the ratio of ๐:๐ is 3: 2, answer each question below.
a. Write an equation for ๐๐ in terms of ๐๐.
b. Write an equation for ๐๐ in terms of ๐๐.
c. Use one of your equations to find ๐1 in terms of ๐ if ๐1 = 1.2(๐).
d. What is the relationship between ๐1 and ๐?
e. What constant, ๐, relates ๐1 and ๐? Is this surprising? Why or why not?f. Using the formula ๐๐ = ๐ โ ๐๐โ1, find ๐3 in terms of ๐.
g. Using the formula ๐๐ = ๐ โ ๐๐โ1, find ๐3 in terms of ๐.
h. Use your answers from parts (f) and (g) to calculate the value of the ratio of ๐3: ๐3?
5. Julius wants to try to estimate the circumference of the earth based on measurements made near his home. Hecannot find a location near his home where the sun is straight overhead. Will he be able to calculate thecircumference of the earth? If so, explain and draw a diagram to support your claim.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 20 GEOMETRY
Lesson 20: How Far Away Is the Moon? Date: 9/26/14 S.130
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Lesson 20: How Far Away Is the Moon?
Classwork
Opening Exercise
What is a solar eclipse? What is a lunar eclipse?
Discussion
Solar Eclipse
3D view:
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 20 GEOMETRY
Lesson 20: How Far Away Is the Moon? Date: 9/26/14 S.131
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Lunar Eclipse
3D view:
Example
a. If the circumference of the earth is about 25,000 miles, what is the Earthโs diameter in miles?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 20 GEOMETRY
Lesson 20: How Far Away Is the Moon? Date: 9/26/14 S.132
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b. Using (a), what is the moonโs diameter in miles?
c. How far away is the moon in miles?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 20 GEOMETRY
Lesson 20: How Far Away Is the Moon? Date: 9/26/14 S.133
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Problem Set
1. If the sun and the moon do not have the same diameter, explain how the sunโs light can be covered by the moonduring a solar eclipse.
2. What would a lunar eclipse look like when viewed from the moon?
3. Suppose you live on a planet with a moon, where during a solar eclipse, the moon appears to be half the diameter ofthe sun.
a. Draw a diagram of how the moon would look against the sun during a solar eclipse.
b. A 1 inch diameter marble held 100 inches away on the planet barely blocks the sun. How many moondiameters away is the moon from the planet? Draw and label a diagram to support your answer.
c. If the diameter of the moon is approximately 35 of the diameter of the planet, and the circumference of the
planet is 185,000 miles, approximately how far is the moon from the planet?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 GEOMETRY
Lesson 21: Special Relationships Within Right TrianglesโDividing into Two Similar Sub-Triangles
Date: 9/26/14 S.134
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Lesson 21: Special Relationships Within Right TrianglesโDividing
into Two Similar Sub-Triangles
Classwork
Opening Exercise
Use the diagram below to complete parts (a)โ(c).
a. Are the triangles shown above similar? Explain.
b. Determine the unknown lengths of the triangles.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 GEOMETRY
Lesson 21: Special Relationships Within Right TrianglesโDividing into Two Similar Sub-Triangles
Date: 9/26/14 S.135
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c. Explain how you found the lengths in part (a).
Example 1
Recall that an altitude of a triangle is a perpendicular line segment from a vertex to the line determined by the opposite side. In triangle โณ ๐ด๐ต๐ถ below, ๐ต๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is the altitude from vertex ๐ต to the line containing ๐ด๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
How many triangles do you see in the figure?
Identify the three triangles by name.
Note that there are many ways to name the three triangles. Ensure that the names students give show corresponding angles.
We want to consider the altitude of a right triangle from the right angle to the hypotenuse. The altitude of a right triangle splits the triangle into two right triangles, each of which shares a common acute angle with the original triangle. In โณ ๐ด๐ต๐ถ, the altitude ๐ต๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ divides the right triangle into two sub-triangles, โณ ๐ต๐ท๐ถ and โณ ๐ด๐ท๐ต.
Is โณ ๐ด๐ต๐ถ~ โณ ๐ต๐ท๐ถ? Is โณ ๐ด๐ต๐ถ~ โณ ๐ด๐ท๐ต? Explain.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 GEOMETRY
Lesson 21: Special Relationships Within Right TrianglesโDividing into Two Similar Sub-Triangles
Date: 9/26/14 S.136
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Is โณ ๐ด๐ต๐ถ~ โณ ๐ท๐ต๐ถ? Explain.
Since โณ ๐ด๐ต๐ถ ~ โณ ๐ต๐ท๐ถ and โณ ๐ด๐ต๐ถ~ โณ ๐ด๐ท๐ต, can we conclude that โณ ๐ต๐ท๐ถ~ โณ ๐ด๐ท๐ต? Explain.
Identify the altitude drawn in triangle โณ ๐ธ๐น๐บ.
As before, the altitude divides the triangle into three triangles. Identify them by name so that the corresponding angles match up.
Does the altitude divide โณ ๐ธ๐น๐บ into three similar sub-triangles as the altitude did with โณ ๐ด๐ต๐ถ?
The fact that the altitude drawn from the right angle of a right triangle divides the triangle into two similar sub-triangles which are also similar to the original triangle allows us to determine the unknown lengths of right triangles.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 GEOMETRY
Lesson 21: Special Relationships Within Right TrianglesโDividing into Two Similar Sub-Triangles
Date: 9/26/14 S.137
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Example 2
Consider the right triangle โณ ๐ด๐ต๐ถ below.
Draw the altitude ๐ต๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ from vertex ๐ต to the line containing ๐ด๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Label the segment ๐ด๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ as ๐ฅ, the segment ๐ท๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ as ๐ฆ, and the segment ๐ต๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ as ๐ง.
Find the values of ๐ฅ,๐ฆ, and ๐ง.
Now we will look at a different strategy for determining the lengths of ๐ฅ,๐ฆ, and ๐ง. The strategy requires that we complete a table of ratios that compares different parts of each triangle.
Make a table of ratios for each triangle that relates the sides listed in the column headers.
๐ฌ๐ก๐จ๐ซ๐ญ๐๐ซ ๐ฅ๐๐ :๐ก๐ฒ๐ฉ๐จ๐ญ๐๐ง๐ฎ๐ฌ๐ ๐ฅ๐จ๐ง๐ ๐๐ซ ๐ฅ๐๐ :๐ก๐ฒ๐ฉ๐จ๐ญ๐๐ง๐ฎ๐ฌ๐ ๐ฌ๐ก๐จ๐ซ๐ญ๐๐ซ ๐ฅ๐๐ : ๐ฅ๐จ๐ง๐ ๐๐ซ ๐ฅ๐๐
โณ ๐จ๐ฉ๐ช
โณ ๐จ๐ซ๐ฉ
โณ ๐ช๐ซ๐ฉ
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 GEOMETRY
Lesson 21: Special Relationships Within Right TrianglesโDividing into Two Similar Sub-Triangles
Date: 9/26/14 S.138
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Our work in Example 1 showed us that โณ ๐ด๐ต๐ถ~ โณ ๐ด๐ท๐ต~ โณ ๐ถ๐ท๐ต. Since the triangles are similar, the ratios of their corresponding sides will be equal. For example, we can find the length of ๐ฅ by equating the values of ๐ โ๐๐๐ก๐๐ ๐๐๐: โ๐ฆ๐๐๐ก๐๐๐ข๐ ๐ ratios of triangles โณ ๐ด๐ต๐ถ and โณ ๐ด๐ท๐ต.
๐ฅ5
=5
1313๐ฅ = 25
๐ฅ =2513
= 11213
Why can we use these ratios to determine the length of ๐ฅ?
Which ratios can we use to determine the length of ๐ฆ?
Use ratios to determine the length of ๐ง.
Since corresponding ratios within similar triangles are equal, we can solve for any unknown side length by equating the values of the corresponding ratios. In the coming lessons, we will learn about more useful ratios for determining unknown side lengths of right triangles.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 GEOMETRY
Lesson 21: Special Relationships Within Right TrianglesโDividing into Two Similar Sub-Triangles
Date: 9/26/14 S.139
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Problem Set
1. Use similar triangles to find the length of the altitudes labeled with variables in each triangle below.
a.
b.
c.
d. Describe the pattern that you see in your calculations for parts (a) through (c).
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 GEOMETRY
Lesson 21: Special Relationships Within Right TrianglesโDividing into Two Similar Sub-Triangles
Date: 9/26/14 S.140
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2. Given right triangle ๐ธ๐น๐บ with altitude ๐น๐ป๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ drawn to thehypotenuse, find the lengths of ๐ธ๐ป, ๐น๐ป, and ๐บ๐ป.
3. Given triangle ๐ผ๐๐ฝ with altitude ๐ฝ๐ฟ๏ฟฝ , ๐ฝ๐ฟ = 32, and ๐ผ๐ฟ = 24, find ๐ผ๐ฝ, ๐ฝ๐, ๐ฟ๐, and ๐ผ๐.
4. Given right triangle ๐ ๐๐ with altitude ๐ ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to its hypotenuse, ๐๐ = 1 2425
, and ๐ ๐ = 6 1825
, find the lengths of the sides of โณ ๐ ๐๐.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 GEOMETRY
Lesson 21: Special Relationships Within Right TrianglesโDividing into Two Similar Sub-Triangles
Date: 9/26/14 S.141
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5. Given right triangle ๐ด๐ต๐ถ with altitude ๐ถ๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, find ๐ด๐ท, ๐ต๐ท, ๐ด๐ต, and ๐ท๐ถ.
6. Right triangle ๐ท๐ธ๐ถ is inscribed in a circle with radius ๐ด๐ถ = 5. ๐ท๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ is a diameter of the circle, ๐ธ๐น is an altitude ofโณ ๐ท๐ธ๐ถ, and ๐ท๐ธ = 6. Find the lengths ๐ฅ and ๐ฆ.
7. In right triangle ๐ด๐ต๐ท, ๐ด๐ต = 53, and altitude ๐ท๐ถ = 14. Find the lengths of ๐ต๐ถ and ๐ด๐ถ.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 22 GEOMETRY
Lesson 22: Multiplying and Dividing Expressions with Radicals Date: 9/26/14 S.142
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Lesson 22: Multiplying and Dividing Expressions with Radicals
Classwork
Exercises 1โ5
Simplify as much as possible.
1. โ172 =
2. โ510 =
3. โ4๐ฅ4 =
4. Complete parts (a) through (c).
a. Compare the value of โ36 to the value of โ9 ร โ4.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 22 GEOMETRY
Lesson 22: Multiplying and Dividing Expressions with Radicals Date: 9/26/14 S.143
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b. Make a conjecture about the validity of the following statement. For nonnegative real numbers ๐ and ๐,โ๐๐ = โ๐ โ โ๐. Explain.
c. Does your conjecture hold true for ๐ = โ4 and ๐ = โ9?
5. Complete parts (a) through (c).
a. Compare the value of ๏ฟฝ10025
to the value of โ100โ25
.
b. Make a conjecture about the validity of the following statement. For nonnegative real numbers ๐ and ๐, when
๐ โ 0, ๏ฟฝ๐๐ = โ๐
โ๐. Explain.
c. Does your conjecture hold true for ๐ = โ100 and ๐ = โ25?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 22 GEOMETRY
Lesson 22: Multiplying and Dividing Expressions with Radicals Date: 9/26/14 S.144
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Exercises 6โ17
Simplify each expression as much as possible and rationalize denominators when applicable.
6. โ72 = 7. ๏ฟฝ1725
=
8. โ32๐ฅ = 9. ๏ฟฝ13
=
10. โ54๐ฅ2 = 11. โ36โ18
=
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 22 GEOMETRY
Lesson 22: Multiplying and Dividing Expressions with Radicals Date: 9/26/14 S.145
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12. ๏ฟฝ 4๐ฅ4
= 13. 4๐ฅ๏ฟฝ64๐ฅ2
=
14. 5๏ฟฝ๐ฅ7
= 15. ๏ฟฝ๐ฅ5
2=
16. โ18๐ฅ3๏ฟฝ๐ฅ5
=
![Page 146: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/146.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 22 GEOMETRY
Lesson 22: Multiplying and Dividing Expressions with Radicals Date: 9/26/14 S.146
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17. The captain of a ship recorded the shipโs coordinates, then sailed north and then west, and then recorded the newcoordinates. The coordinates were used to calculate the distance they traveled, โ578 km. When the captain askedhow far they traveled, the navigator said, โAbout 24 km.โ Is the navigator correct? Under what conditions might heneed to be more precise in his answer?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 22 GEOMETRY
Lesson 22: Multiplying and Dividing Expressions with Radicals Date: 9/26/14 S.147
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Problem Set
Express each number in its simplest radical form.
1. โ6 โ โ60 = 2. โ108 =
3. Pablo found the length of the hypotenuse of a right triangle to be โ45. Can the length be simplified? Explain.
4. โ12๐ฅ4 =
5. Sarahi found the distance between two points on a coordinate plane to be โ74. Can this answer be simplified?Explain.
6. โ16๐ฅ3 =
7. โ27โ3
=
8. Nazem and Joffrey are arguing about who got the right answer. Nazem says the answer is 1โ3
, and Joffrey says the
answer is โ33
. Show and explain that their answers are equivalent.
9. ๏ฟฝ58
=
10. Determine the area of a square with side length 2โ7 in.
11. Determine the exact area of the shaded region shown below.
![Page 148: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/148.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 22 GEOMETRY
Lesson 22: Multiplying and Dividing Expressions with Radicals Date: 9/26/14 S.148
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12. Determine the exact area of the shaded region shown to the right.
13. Calculate the area of the triangle to the right.
14. ๏ฟฝ2๐ฅ3โ โ8๐ฅ๏ฟฝ๐ฅ3
=
15. Prove Rule 2 for square roots: ๏ฟฝ๐๐
= โ๐โ๐
(๐ โฅ 0, ๐ > 0).
Let ๐ be the nonnegative number so that ๐2 = ๐, and let ๐ be the nonnegative number so that ๐2 = ๐.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 23 GEOMETRY
Lesson 23: Adding and Subtracting Expressions with Radicals Date: 9/26/14 S.149
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Lesson 23: Adding and Subtracting Expressions with Radicals
Classwork
Exercises 1โ5
Simplify each expression as much as possible.
1. โ32 = 2. โ45 =
3. โ300 =
4. The triangle shown below has a perimeter of 6.5โ2 units. Make a conjecture about how this answer was reached.
5. The sides of a triangle are 4โ3, โ12, and โ75. Make a conjecture about how to determine the perimeter of thistriangle.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 23 GEOMETRY
Lesson 23: Adding and Subtracting Expressions with Radicals Date: 9/26/14 S.150
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Exercise 6
6. Circle the expressions that can be simplified using the distributive property. Be prepared to explain your choices.
8.3โ2 + 7.9โ2
โ13 โ โ6
โ15โ5 + โ45
11โ7 โ 6โ7 + 3โ2
19โ2 + 2โ8
4 + โ11
โ7 + 2โ10
โ12 โ โ75
โ32 + โ2
6โ13 + โ26
Example 1
Explain how the expression 8.3โ2 + 7.9โ2 can be simplified using the distributive property.
Explain how the expression 11โ7 โ 6โ7 + 3โ2 can be simplified using the distributive property.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 23 GEOMETRY
Lesson 23: Adding and Subtracting Expressions with Radicals Date: 9/26/14 S.151
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Example 2
Explain how the expression 19โ2 + 2โ8 can be simplified using the distributive property.
Example 3
Can the expression โ7 + 2โ10 be simplified using the distributive property?
To determine if an expression can be simplified, you must first simplify each of the terms within the expression. Then, apply the distributive property, or other properties as needed, to simplify the expression.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 23 GEOMETRY
Lesson 23: Adding and Subtracting Expressions with Radicals Date: 9/26/14 S.152
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Problem Set
Express each answer in simplified radical form.
1. 18โ5 โ 12โ5 = 2. โ24 + 4โ54 =
3. 2โ7 + 4โ63 =
4. What is the perimeter of the triangle shown below?
5. Determine the area and perimeter of the triangle shown. Simplify as much as possible.
6. Determine the area and perimeter of the rectangle shown. Simplify as much as possible.
7. Determine the area and perimeter of the triangle shown. Simplify as much as possible.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 23 GEOMETRY
Lesson 23: Adding and Subtracting Expressions with Radicals Date: 9/26/14 S.153
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8. Determine the area and perimeter of the triangle shown. Simplify as much as possible.
9. The area of the rectangle shown in the diagram below is 160 square units. Determine the area and perimeter of theshaded triangle. Write your answers in simplest radical form, and then approximate to the nearest tenth.
10. Parallelogram ๐ด๐ต๐ถ๐ท has an area of 9โ3. ๐ท๐ถ = 3โ3, and ๐บ and ๐ป are midpoints of ๐ท๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ถ๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, respectively. Findthe area of the shaded region. Write your answer in simplest radical form.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 24 GEOMETRY
Lesson 24: Prove Pythagorean Theorem Using Similarity Date: 9/26/14 S.154
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Lesson 24: Prove the Pythagorean Theorem Using Similarity
Classwork
Exercises 1โ3
1. Find the length of the hypotenuse of a right triangle whose legs have lengths 50 and 100.
2. Can you think of a simpler method for finding the length of the hypotenuse in Exercise 1? Explain.
3. Find the length of the hypotenuse of a right triangle whose legs have lengths 75 and 225.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 24 GEOMETRY
Lesson 24: Prove Pythagorean Theorem Using Similarity Date: 9/26/14 S.155
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Exploratory Challenge/Exercises 4โ5
4. An equilateral triangle has sides of length 2 and angle measures of 60, as shown below. The altitude from onevertex to the opposite side divides the triangle into two right triangles.
a. Are those triangles congruent? Explain.
b. What is the length of the shorter leg of each of the right triangles? Explain.
c. Use the Pythagorean theorem to determine the length of the altitude.
d. Write the ratio that represents ๐ โ๐๐๐ก๐๐ ๐๐๐: โ๐ฆ๐๐๐ก๐๐๐ข๐ ๐.
![Page 156: Lesson 1: Scale Drawings...dilation is. Example 1 Create a scale drawing of the figure below using the ratio method about center ๐ and scale factor ๐= 1 2. Step 1. Draw a ray](https://reader034.fdocuments.in/reader034/viewer/2022042808/5f8a76c53b1abd39df3ff525/html5/thumbnails/156.jpg)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 24 GEOMETRY
Lesson 24: Prove Pythagorean Theorem Using Similarity Date: 9/26/14 S.156
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e. Write the ratio that represents ๐๐๐๐๐๐ ๐๐๐: โ๐ฆ๐๐๐ก๐๐๐ข๐ ๐.
f. Write the ratio that represents ๐ โ๐๐๐ก๐๐ ๐๐๐: ๐๐๐๐๐๐ ๐๐๐.
g. By the AA criterion, any triangles with measures 30โ60โ90 will be similar to this triangle. If a 30โ60โ90triangle has a hypotenuse of length 16, what are the lengths of the legs?
5. An isosceles right triangle has leg lengths of 1, as shown.
a. What are the measures of the other two angles? Explain.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 24 GEOMETRY
Lesson 24: Prove Pythagorean Theorem Using Similarity Date: 9/26/14 S.157
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b. Use the Pythagorean theorem to determine the length of the hypotenuse of the right triangle.
c. Is it necessary to write all three ratios: ๐ โ๐๐๐ก๐๐ ๐๐๐: โ๐ฆ๐๐๐ก๐๐๐ข๐ ๐, ๐๐๐๐๐๐ ๐๐๐: โ๐ฆ๐๐๐ก๐๐๐ข๐ ๐, and๐ โ๐๐๐ก๐๐ ๐๐๐: ๐๐๐๐๐๐ ๐๐๐? Explain.
d. Write the ratio that represents ๐๐๐:โ๐ฆ๐๐๐ก๐๐๐ข๐ ๐.
e. By the AA criterion, any triangles with measures 45โ45โ90 will be similar to this triangle. If a 45โ45โ90triangle has a hypotenuse of length 20, what are the lengths of the legs?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 24 GEOMETRY
Lesson 24: Prove Pythagorean Theorem Using Similarity Date: 9/26/14 S.158
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Problem Set
1. In each row of the table below are the lengths of the legs and hypotenuses of different right triangles. Find themissing side lengths in each row, in simplest radical form.
Leg1 Leg2 Hypotenuse 15 25 15 36 3 7
100 200
2. Claude sailed his boat due south for 38 miles, then due west for 25 miles. Approximately how far is Claude fromwhere he began?
3. Find the lengths of the legs in the triangle given the hypotenuse with length 100.
4. Find the length of the hypotenuse in the right triangle given that the legshave lengths of 100.
5. Each row in the table below shows the side lengths of a different 30โ60โ90 right triangle. Complete the table withthe missing side lengths in simplest radical form. Use the relationships of the values in the first three rows tocomplete the last row. How could the expressions in the last row be used?
Shorter Leg Longer Leg Hypotenuse 25 50 15
3 2โ3 ๐ฅ
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 24 GEOMETRY
Lesson 24: Prove Pythagorean Theorem Using Similarity Date: 9/26/14 S.159
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6. In right triangle ๐ด๐ต๐ถ with โ ๐ถ a right angle, an altitude of length โ is dropped to side ๐ด๐ต that splits the side ๐ด๐ต intosegments of length ๐ฅ and ๐ฆ. Use the Pythagorean theorem to show โ2 = ๐ฅ๐ฆ.
7. In triangle ๐ด๐ต๐ถ, the altitude from โ ๐ถ splits side ๐ด๐ต into two segments of lengths ๐ฅ and ๐ฆ. If โ denotes the length ofthe altitude and โ2 = ๐ฅ๐ฆ, use the Pythagorean theorem and its converse to show that triangle ๐ด๐ต๐ถ is a righttriangle with โ ๐ถ a right angle.
.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 25 GEOMETRY
Lesson 25: Incredibly Useful Ratios Date: 9/26/14 S.160
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Lesson 25: Incredibly Useful Ratios
Classwork
Exercises 1โ3
Use the right triangle โณ ๐ด๐ต๐ถ to answer Exercises 1โ3.
1. Name the side of the triangle opposite โ ๐ด.
2. Name the side of the triangle opposite โ ๐ต.
3. Name the side of the triangle opposite โ ๐ถ.
Exercises 4โ6
For each exercise, label the appropriate sides as adjacent, opposite, and hypotenuse, with respect to the marked acute angle.
4.
5.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 25 GEOMETRY
Lesson 25: Incredibly Useful Ratios Date: 9/26/14 S.161
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6.
Exploratory Challenge
Note: Angle measures are approximations.
For each triangle in your set, determine missing angle measurements and side lengths. Side lengths should be measured to one decimal place. Make sure that each of the ๐๐๐
โ๐ฆ๐ and ๐๐๐
โ๐ฆ๐ ratios are set up and missing values are calculated to one
decimal place.
Group 1
Triangle Angle Measures Length Measures ๐๐๐๐๐๐
๐๐ ๐๐๐๐
1. โณ ๐ด๐ต๐ถ 1213
โ 0.92 5
13โ 0.38
2. โณ ๐ท๐ธ๐น ๐โ ๐ท โ 53ห ๐ท๐ธ = 3 cm ๐ธ๐น = 4 cm ๐ท๐น = 5 cm
3. โณ ๐บ๐ป๐ผ ๐โ ๐ผ โ 41ห ๐บ๐ป = 5.3 cm 5.3โ 0.66 = 0.75
4. โณ ๐ฝ๐พ๐ฟ ๐พ๐ฟ = 6.93 cm ๐ฝ๐ฟ = 8 cm 8
= 8โ 0.87
5. โณ๐ด๐ต๐ถ ๐๐.๐
โ ๐.๐๐ ๐.๐๐.๐
โ ๐.๐๐
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 25 GEOMETRY
Lesson 25: Incredibly Useful Ratios Date: 9/26/14 S.162
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Group 2
Triangle Angle Measures Length Measures ๐๐๐๐๐๐
๐๐ ๐๐๐๐
1. โณ ๐ดโฒ๐ตโฒ๐ถโฒ 6
6.5โ 0.92
2.56.5
โ 0.38
2. โณ ๐ทโฒ๐ธโฒ๐นโฒ ๐โ ๐ทโฒ โ 53ห ๐ทโฒ๐ธโฒ = 6 cm ๐ธโฒ๐นโฒ = 8 cm ๐ทโฒ๐นโฒ = 10 cm
3. โณ ๐บโฒ๐ปโฒ๐ผโฒ ๐โ ๐ผโฒ โ 41ห ๐บโฒ๐ปโฒ = 7.9 cm 7.9โ 0.66 = 0.75
4. โณ ๐ฝโฒ๐พโฒ๐ฟโฒ ๐พโฒ๐ฟโฒ = 10.4 cm ๐ฝโฒ๐ฟโฒ = 12 cm 12
= 12
โ 0.87
5. โณ๐โฒ๐โฒ๐โฒ 8
17โ 0.47
1517
โ 0.88
With a partner, discuss what you can conclude about each pair of triangles between the two sets?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 25 GEOMETRY
Lesson 25: Incredibly Useful Ratios Date: 9/26/14 S.163
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Exercises 7โ10
For each question, approximate the unknown lengths to one decimal place. Refer back to your completed chart from the Exploratory Challenge; each indicated acute angle is the same approximated acute angle measure as in the chart. Set up and label the appropriate length ratios, using the terms ๐๐๐, ๐๐๐, and โ๐ฆ๐ in the set up of each ratio.
7.
8.
9.
10. From a point 120 m away from a building, Serena measures the angle between the ground and the top of a buildingand finds it measures 41ห.What is the height of the building? Round to the nearest meter.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 25 GEOMETRY
Lesson 25: Incredibly Useful Ratios Date: 9/26/14 S.164
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Problem Set
The table below contains the values of the ratios ๐๐๐โ๐ฆ๐
and ๐๐๐โ๐ฆ๐
for a variety of right triangles based on a given acute angle,
๐, from each triangle. Use the table and the diagram of the right triangle below to complete each problem.
๐ (degrees) 0 10 20 30 40 45 50 60 70 80 90
๐๐๐โ๐ฆ๐ 0 0.1736 0.3420
12
= 0.5 0.6428 0.7071 0.7660 0.8660 0.9397 0.9848 1
๐๐๐โ๐ฆ๐
1 0.9848 0.9397 0.8660 0.7660 0.7071 0.6428 12
= 0.5 0.3420 0.1736 0
For each problem, approximate the unknown lengths to one decimal place. Write the appropriate length ratios, using the terms ๐๐๐, ๐๐๐, and โ๐ฆ๐ in the set up of each ratio.
1. Find the approximate length of the leg opposite the 80ยฐ angle.
2. Find the approximate length of the hypotenuse.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 25 GEOMETRY
Lesson 25: Incredibly Useful Ratios Date: 9/26/14 S.165
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3. Find the approximate length of the hypotenuse.
4. Find the approximate length of the leg adjacent to the 40ยฐ angle.
5. Find the length of both legs of the right triangle below. Indicate which leg is adjacent and which is opposite thegiven angle of 30ยฐ.
U
6. Three city streets form a right triangle. Main Street and State Street are perpendicular. Laura Street and StateStreet intersect at a 50ยฐ angle. The distance along Laura Street to Main Street is 0.8 mile. If Laura Street is closedbetween Main Street and State Street for a festival, approximately how far (to the nearest tenth) will someone haveto travel to get around the festival if they take only Main Street and State Street?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 25 GEOMETRY
Lesson 25: Incredibly Useful Ratios Date: 9/26/14 S.166
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7. A cable anchors a utility pole to the ground as shown in the picture. The cable forms an angle of 70ยฐ with theground. The distance from the base of the utility pole to the anchor point on the ground is 3.8 meters.Approximately how long is the support cable?
8. Indy says that the ratio of ๐๐๐๐๐๐
for an angle of 0ยฐ has a value of 0 because the opposite side of the triangle has a
length of 0. What does she mean?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 26 GEOMETRY
Lesson 26: Definition of Sine, Cosine, and Tangent Date: 9/26/14 S.167
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Lesson 26: Definition of Sine, Cosine, and Tangent
Classwork
Exercises 1โ3
1. Identify the ๐๐๐โ๐ฆ๐
ratios for angles โ ๐ด and โ ๐ต.
2. Identify the ๐๐๐โ๐ฆ๐
ratios for angles โ ๐ด and โ ๐ต.
3. Describe the relationship between the ratios for angles โ ๐ด and โ ๐ต.
Exercises 4โ9
4. In โณ ๐๐๐ , ๐โ ๐ = 53.2ยฐ and ๐โ ๐ = 36.8ยฐ. Complete the following table.
Measure of Angle Sine ๏ฟฝ๐๐๐โ๐ฆ๐
๏ฟฝ Cosine ๏ฟฝ๐๐๐โ๐ฆ๐
๏ฟฝ Tangent ๏ฟฝ๐๐๐๐๐๐
๏ฟฝ
53.2
36.8
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 26 GEOMETRY
Lesson 26: Definition of Sine, Cosine, and Tangent Date: 9/26/14 S.168
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5. In the triangle below, ๐โ ๐ด = 33.7ยฐ and ๐โ ๐ต = 56.3ยฐ. Complete the following table.
Measure of Angle Sine Cosine Tangent
33.7
56.3
6. In the triangle below, let ๐ be the measure of โ ๐ธ and ๐ be the measure of โ ๐ท. Complete the following table.
Measure of Angle Sine Cosine Tangent
๐
๐
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 26 GEOMETRY
Lesson 26: Definition of Sine, Cosine, and Tangent Date: 9/26/14 S.169
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7. In the triangle below, let ๐ฅ be the measure of โ ๐ and ๐ฆ be the measure of โ ๐. Complete the following table.
Measure of Angle Sine Cosine Tangent
๐ฅ
๐ฆ
8. Tamer did not finish completing the table below for a diagram similar to the previous problems that the teacher hadon the board where ๐ was the measure of โ ๐ and ๐ was the measure of โ ๐. Use any patterns you notice fromExercises 1โ4 to complete the table for Tamer.
Measure of Angle Sine Cosine Tangent
๐ sin๐ =11โ157
cos ๐ =6
โ157tan๐ =
116
๐
9. Explain how you were able to determine the sine, cosine, and tangent of โ ๐ in Exercise 7.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 26 GEOMETRY
Lesson 26: Definition of Sine, Cosine, and Tangent Date: 9/26/14 S.170
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Problem Set
1. Given the triangle in the diagram, complete the following table.
Angle Measure sin cos tan
๐ผ
๐ฝ
2. Given the table of values below (not in simplest radical form), label the sides and angles in the right triangle.
Angle Measure sin cos tan
๐ผ 4
2โ102โ6
2โ10
42โ6
๐ฝ 2โ6
2โ10
42โ10
2โ64
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 26 GEOMETRY
Lesson 26: Definition of Sine, Cosine, and Tangent Date: 9/26/14 S.171
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3. Given sin ๐ผ and sin ๐ฝ, complete the missing values in the table. You may draw a diagram to help you.
Angle Measure sin cos tan
๐ผ โ23โ3
53โ3
๐ฝ
4. Given the triangle shown to the right, fill in the missing values in thetable.
Angle Measure sin cos tan
๐ผ
๐ฝ
5. Jules thinks that if ๐ผ and ๐ฝ are two different acute angle measures, then sin๐ผ โ sin๐ฝ. Do you agree or disagree?Explain.
6. Given the triangle in the diagram, complete the following table.
Angle Measure sin cos tan
๐ผ
๐ฝ
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 26 GEOMETRY
Lesson 26: Definition of Sine, Cosine, and Tangent Date: 9/26/14 S.172
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Rewrite the values from the table in simplest terms.
Angle Measure sin cos tan
๐ผ
๐ฝ
Draw and label the sides and angles of a right triangle using the values of the ratios sin and cos. How is the new triangle related to the original triangle?
7. Given tan ๐ผ and cos ๐ฝ, in simplest terms, find the missing side lengths of the right triangle if one leg of the trianglehas a length of 4. Draw and label the sides and angles of the right triangle.
Angle sin ๐ cos ๐ tan ๐
๐ผ
๐ฝ
8. Eric wants to hang a rope bridge over a small ravine so that it is easier to cross. To hang the bridge, he needs toknow how much rope is needed to span the distance between two trees that are directly across from each other oneither side of the ravine. Help Eric devise a plan using sine, cosine, and tangent to determine the approximatedistance from tree A to tree B without having to cross the ravine.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 26 GEOMETRY
Lesson 26: Definition of Sine, Cosine, and Tangent Date: 9/26/14 S.173
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9. A fisherman is at point ๐น on the open sea and has three favorite fishing locations. The locations are indicated bypoints ๐ด,๐ต, and ๐ถ. The fisherman plans to sail from ๐น to ๐ด, then to ๐ต, then to ๐ถ, then back to ๐น. If the fisherman is14 miles from ๐ด๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ, find the total distance that he will sail.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 27 GEOMETRY
Lesson 27: Sine and Cosine of Complementary and Special Angles Date: 9/26/14 S.174
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Lesson 27: Sine and Cosine of Complementary and Special Angles
Classwork
Example 1
If ๐ผ and ๐ฝ are the measurements of complementary angles, then we are going to show that sin ๐ผ = cos ๐ฝ.
In right triangle ๐ด๐ต๐ถ, the measurement of acute angle โ ๐ด is denoted by ๐ผ, and the measurement of acute angle โ ๐ต is denoted by ๐ฝ.
Determine the following values in the table:
sin ๐ผ sin ๐ฝ cos ๐ผ cos ๐ฝ
What can you conclude from the results?
Exercises 1โ3
1. Consider the right triangle ๐ด๐ต๐ถ so that โ ๐ถ is a right angle, and the degree measures of โ ๐ด and โ ๐ต are ๐ผ and ๐ฝ,respectively.
a. Find ๐ผ + ๐ฝ.
b. Use trigonometric ratios to describe ๐ต๐ถ๐ด๐ต
two different ways.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 27 GEOMETRY
Lesson 27: Sine and Cosine of Complementary and Special Angles Date: 9/26/14 S.175
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c. Use trigonometric ratios to describe ๐ด๐ถ๐ด๐ต
two different ways.
d. What can you conclude about sin ๐ผ and cos ๐ฝ?
e. What can you conclude about cos ๐ผ and sin ๐ฝ?
2. Find values for ๐ that make each statement true.a. sin ๐ = cos (25)
b. sin 80 = cos ๐
c. sin ๐ = cos (๐ + 10)
d. sin (๐ โ 45) = cos (๐)
3. For what angle measurement must sine and cosine have the same value? Explain how you know.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 27 GEOMETRY
Lesson 27: Sine and Cosine of Complementary and Special Angles Date: 9/26/14 S.176
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Example 2
What is happening to ๐ and ๐ as ๐ changes? What happens to sin๐ and cos ๐?
Example 3
There are certain special angles where it is possible to give the exact value of sine and cosine. These are the angles that measure 0ห, 30ห, 45ห, 60ห, and 90ห; these angle measures are frequently seen.
You should memorize the sine and cosine of these angles with quick recall just as you did your arithmetic facts.
a. Learn the following sine and cosine values of the key angle measurements.
๐ 0ห 30ห 45ห 60ห 90ห
Sine 0 12
โ22
โ32
1
Cosine 1 โ32
โ22
12
0
We focus on an easy way to remember the entries in the table. What do you notice about the table values?
This is easily explained because the pairs (0,90), (30,60), and (45,45) are the measures of complementary angles. So, for instance, sin 30 = cos 60.
The sequence 0, 12
, โ22
, โ32
, 1 may be easier to remember as the sequence โ02
, โ12
, โ22
, โ32
, โ42
.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 27 GEOMETRY
Lesson 27: Sine and Cosine of Complementary and Special Angles Date: 9/26/14 S.177
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b. โณ ๐ด๐ต๐ถ is equilateral, with side length 2; ๐ท is the midpoint of side ๐ด๐ถ. Label all side lengths and anglemeasurements for โณ ๐ด๐ต๐ท. Use your figure to determine the sine and cosine of 30 and 60.
c. Draw an isosceles right triangle with legs of length 1. What are the measures of the acute angles of thetriangle? What is the length of the hypotenuse? Use your triangle to determine sine and cosine of the acuteangles.
Parts (b) and (c) demonstrate how the sine and cosine values of the mentioned special angles can be found. These triangles are common to trigonometry; we refer to the triangle in part (b) as a 30โ60โ90 triangle and the triangle in part (c) as a 45โ45โ90 triangle.
30โ60โ90 Triangle, side length ratio 1: 2:โ3 45โ45โ90 Triangle, side length ratio 1: 1:โ2
2: 4: 2โ3 2: 2: 2โ2
3: 6: 3โ3 3: 3: 3โ2
4: 8: 4โ3 4: 4: 4โ2
๐ฅ: 2๐ฅ: ๐ฅโ3 ๐ฅ: ๐ฅ: ๐ฅโ2
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 27 GEOMETRY
Lesson 27: Sine and Cosine of Complementary and Special Angles Date: 9/26/14 S.178
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Exercises 4โ5
4. Find the missing side lengths in the triangle.
5. Find the missing side lengths in the triangle.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 27 GEOMETRY
Lesson 27: Sine and Cosine of Complementary and Special Angles Date: 9/26/14 S.179
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Problem Set
1. Find the value of ๐ that makes each statement true.a. sin๐ = cos(๐ + 38)b. cos ๐ = sin(๐ โ 30)c. sin๐ = cos(3๐ + 20)
d. sin ๏ฟฝ๐3
+ 10๏ฟฝ = cos๐
2. a. Make a prediction about how the sum sin 30 + cos 60 will relate to the sum sin 60 + cos 30.
b. Use the sine and cosine values of special angles to find the sum: sin 30 + cos 60.c. Find the sum sin 60 + cos 30.
d. Was your prediction a valid prediction? Explain why or why not.
3. Langdon thinks that the sum sin 30 + sin 30 is equal to sin 60. Do you agree with Langdon? Explain what thismeans about the sum of the sines of angles.
4. A square has side lengths of 7โ2. Use sine or cosine to find the length of the diagonal of the square. Confirm youranswer using the Pythagorean theorem.
5. Given an equilateral triangle with sides of length 9, find the length of the altitude. Confirm your answer using thePythagorean theorem.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 28 GEOMETRY
Lesson 28: Solving Problems Using Sine and Cosine Date: 9/26/14 S.180
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Lesson 28: Solving Problems Using Sine and Cosine
Classwork
Exercise 1โ4
1. a. The bus drops you off at the corner of H Street and 1st Street, approximately 300 ft. from school. You plan to
walk to your friend Jannethโs house after school to work on a project. Approximately how many feet will youhave to walk from school to Jannethโs house? Round your answer to the nearest foot. (Hint: Use the ratiosyou developed in Lesson 25.)
b. In real life, it is unlikely that you would calculate the distance between school and Jannethโs house in thismanner. Describe a similar situation in which you might actually want to determine the distance between twopoints using a trigonometric ratio.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 28 GEOMETRY
Lesson 28: Solving Problems Using Sine and Cosine Date: 9/26/14 S.181
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2. Use a calculator to find the sine and cosine of ๐. Give your answer rounded to the ten-thousandth place.
๐ 0 10 20 30 40 50 60 70 80 90
sin๐
cos ๐
3. What do you notice about the numbers in the row sin ๐ compared with the numbers in the row cos ๐?
4. Provide an explanation for what you noticed in Exercise 2.
Example 1
Find the values of ๐ and ๐.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 28 GEOMETRY
Lesson 28: Solving Problems Using Sine and Cosine Date: 9/26/14 S.182
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Exercise 5
5. A shipmate set a boat to sail exactly 27ห NE from the dock. After traveling 120 miles, the shipmate realized he hadmisunderstood the instructions from the captain; he was supposed to set sail going directly east!
a. How many miles will the shipmate have to travel directly south before he is directly east of the dock? Roundyour answer to the nearest mile.
b. How many extra miles does the shipmate travel by going the wrong direction compared to going directly east?Round your answer to the nearest mile.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 28 GEOMETRY
Lesson 28: Solving Problems Using Sine and Cosine Date: 9/26/14 S.183
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Example 2
Johanna borrowed some tools from a friend so that she could precisely, but not exactly, measure the corner space in her backyard to plant some vegetables. She wants to build a fence to prevent her dog from digging up the seeds that she plants. Johanna returned the tools to her friend before making the most important measurement: the one that would give the length of the fence!
Johanna decided that she could just use the Pythagorean theorem to find the length of the fence sheโd need. Is the Pythagorean theorem applicable in this situation? Explain.
Exercise 6
6. The measurements of the triangle shown below are rounded to the nearest hundredth. Calculate the missing sidelength to the nearest hundredth.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 28 GEOMETRY
Lesson 28: Solving Problems Using Sine and Cosine Date: 9/26/14 S.184
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Problem Set
1. Given right triangle ๐บ๐ป๐ผ, with right angle at ๐ป, ๐บ๐ป = 12.2, and ๐โ ๐บ = 28ยฐ, find the measures of the remainingsides and angle to the nearest tenth.
2. The Occupational Safety and Health Administration (OSHA) provides standards for safety at the workplace. A ladderis leaned against a vertical wall according to OSHA standards and forms an angle of approximately 75ยฐ with thefloor.
a. If the ladder is 25 ft. long, what is the distance from the base of the ladder to the base of thewall?
b. How high on the wall does the ladder make contact?c. Describe how to safely set a ladder according to OSHA standards without using a protractor.
3. A regular pentagon with side lengths of 14 cm is inscribed in a circle. What is the radius of the circle?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 28 GEOMETRY
Lesson 28: Solving Problems Using Sine and Cosine Date: 9/26/14 S.185
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Horizontal Center Line
4. The circular frame of a Ferris wheel is suspended so that it sits 4 ft. above the ground and has a radius of 30 ๐๐ก. Asegment joins center ๐ถ to point ๐ on the circle. If ๐ถ๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ makes an angle of 48ยฐ with the horizon, what is the distance ofpoint ๐ to the ground?
5. Tim is a contractor who is designing a wheelchair ramp for handicapped access to a business. According to theAmericans with Disabilities Act (ADA), the maximum slope allowed for a public wheelchair ramp forms an angle ofapproximately 4.76ยฐ to level ground. The length of a rampโs surface cannot exceed 30 ft. without including a flat5 ft. ร 5 ft. platform (minimum dimensions) on which a person can rest, and such a platform must be included at thebottom and top of any ramp.
Tim designs a ramp that forms an angle of 4ยฐ to the level ground to reach the entrance of the building. Theentrance of the building is 2 ft. 9 in. above the ground. Let ๐ฅ and ๐ฆ as shown in Timโs initial design below be theindicated distances in feet.
a. Assuming that the ground in front of the buildingโs entrance is flat, use Timโs measurements and the ADArequirements to complete and/or revise his wheelchair ramp design.
(For more information, see section 405 of the 2010 ADA Standards for Accessible Design at the following link: http://www.ada.gov/regs2010/2010ADAStandards/2010ADAstandards.htm#pgfId-1006877.)
b. What is the total distance from the start of the ramp to the entrance of the building in your design?
6. Tim is designing a roof truss in the shape of an isosceles triangle. The design shows the base angles of the truss tohave measures of 18.5ยฐ. If the horizontal base of the roof truss is 36 ft. across, what is the height of the truss?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 29 GEOMETRY
Lesson 29: Applying Tangents Date: 9/26/14 S.186
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Lesson 29: Applying Tangents
Classwork
Opening Exercise
a. Use a calculator to find the tangent of ๐. Give your answer correct to four decimal places.
๐ 0 10 20 30 40 50 60 70 80 90
sin๐ 0 0.1736 0.3420 0.5 0.6428 0.7660 0.8660 0.9397 0.9848 1
cos ๐ 1 0.9848 0.9397 0.8660 0.7660 0.6428 0.5 0.3420 0.1736 0
sin๐cos ๐tan๐
b. The table from Lesson 29 is provided here for you. In the row labeled ๐ ๐๐ ๐cos ๐
, divide the sine values by the cosine values. What do you notice?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 29 GEOMETRY
Lesson 29: Applying Tangents Date: 9/26/14 S.187
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Example 1
Scott, whose eye level is 1.5 m above the ground, stands 30 m from a tree. The angle of elevation of a bird at the top of the tree is 36ห. How far above ground is the bird?
Example 2
From an angle of depression of 40ห, John watches his friend approach his building while standing on the rooftop. The rooftop is 16 m from the ground, and Johnโs eye level is at about 1.8 m from the rooftop. What is the distance between Johnโs friend and the building?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 29 GEOMETRY
Lesson 29: Applying Tangents Date: 9/26/14 S.188
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Exercise 1
Standing on the gallery of a lighthouse (the deck at the top of a lighthouse), a person spots a ship at an angle of depression of 20ห. The lighthouse is 28 m tall and sits on a cliff 45 m tall as measured from sea level. What is the horizontal distance between the lighthouse and the ship? Sketch a diagram to support your answer.
Exercise 2
A line on the coordinate plane makes an angle of depression of 36ยฐ. Find the slope of the line, correct to four decimal places.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 29 GEOMETRY
Lesson 29: Applying Tangents Date: 9/26/14 S.189
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Problem Set
1. A line in the coordinate plane has an angle of elevation of 53ยฐ. Find the slope of the line correct to four decimalplaces.
2. A line in the coordinate plane has an angle of depression of 25ยฐ. Find the slope of the line correct to four decimalplaces.
3. In Problems 1 and 2, why do the lengths of the legs of the right triangles formed not affect the slope of the line?
4. Given the angles of depression below, determine the slope of the line with the indicated angle correct to fourdecimal places.a. 35ยฐ angle of depression
b. 49ยฐ angle of depression
c. 80ยฐ angle of depression
d. 87ยฐ angle of depression
e. 89ยฐ angle of depression
f. 89.9ยฐ angle of depressiong. What appears to be happening to the slopes (and tangent values) as the angles of depression get closer to
90ยฐ?
h. Find the slopes of angles of depression that are even closer to 90ยฐ than 89.9ยฐ. Can the value of the tangent of90ยฐ be defined? Why or why not?
5. For the indicated angle, express the quotient in terms of sine, cosine, or tangent. Then write the quotient insimplest terms.
a. 42โ13
; ๐ผ
b. 64 ; ๐ผ
c. 42โ13
; ๐ฝ
d. 46 ; ๐ฝ
6. The pitch of a roof on a home is expressed as a ratio of vertical rise: horizontal run where the run has a length of12 units. If a given roof design includes an angle of elevation of 22.5ยฐ, and the roof spans 36 ft. as shown in thediagram, determine the pitch of the roof. Then determine the distance along one of the two sloped surfaces of theroof.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 29 GEOMETRY
Lesson 29: Applying Tangents Date: 9/26/14 S.190
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2 ft.
14 ft.
7. An anchor cable supports a vertical utility pole forming a 51ยฐ angle with the ground. The cable is attached to thetop of the pole. If the distance from the base of the pole to the base of the cable is 5 meters, how tall is the pole?
8. A winch is a tool that rotates a cylinder, around which a cable is wound. When the winch rotates in one direction, itdraws the cable in. Joey is using a winch and a pulley (as shown in the diagram) to raise a heavy box off the floorand onto a cart. The box is 2 ft. tall, and the winch is 14 ft. horizontally from where cable drops down vertically fromthe pulley. The angle of elevation to the pulley is 42ยฐ. What is the approximate length of cable required to connectthe winch and the box?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 30 GEOMETRY
Lesson 30: Trigonometry and the Pythagorean Theorem Date: 9/26/14 S.191
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Figure 1
Lesson 30: Trigonometry and the Pythagorean Theorem
Classwork
Exercises 1โ2
1. In a right triangle, with acute angle of measure ๐, sin๐ = 12. What is the value of cos ๐? Draw a diagram as part of
your response.
2. In a right triangle, with acute angle of measure ๐, sin๐ = 79. What is the value of tan๐? Draw a diagram as part of
your response.
Example 1
a. What common right triangle was probably modeled in the construction of the triangle in Figure 2? Usesin 53 โ 0.8.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 30 GEOMETRY
Lesson 30: Trigonometry and the Pythagorean Theorem Date: 9/26/14 S.192
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b. The actual angle between the base and lateral faces of the pyramid is actually closer to 52ห. Considering theage of the pyramid, what could account for the difference between the angle measure in part (a) and theactual measure?
c. Why do you think the architects chose to use a 3โ4โ5 as a model for the triangle?
Example 2
Show why tan ๐ = sin๐cos๐
.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 30 GEOMETRY
Lesson 30: Trigonometry and the Pythagorean Theorem Date: 9/26/14 S.193
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Exercises 3โ4
3. In a right triangle, with acute angle of measure ๐, sin๐ = 12, use the Pythagorean identity to determine the value of
cos ๐.
4. Given a right triangle, with acute angle of measure ๐, sin๐ = 79, use the Pythagorean identity to determine the value
of tan ๐.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 30 GEOMETRY
Lesson 30: Trigonometry and the Pythagorean Theorem Date: 9/26/14 S.194
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Problem Set
1. If cos ๐ = 45, find sin ๐ and tan๐.
2. If sin ฮธ = 44125
, find cosฮธ and tan ฮธ.
3. If tan ๐ = 5, find sin ๐ and cos ๐.
4. If sin ๐ = โ55
, find cos ๐ and tan ๐.
5. Find the missing side lengths of the following triangle using sine, cosine, and/or tangent.Round your answer to four decimal places.
6. A surveying crew has two points ๐ด and ๐ต marked along a roadside at a distance of 400 yd. A third point ๐ถ is markedat the back corner of a property along a perpendicular to the road at ๐ต. A straight path joining ๐ถ to ๐ด forms a 28ยฐangle with the road. Find the distance from the road to point ๐ถ at the back of the property and the distance from ๐ดto ๐ถ using sine, cosine, and/or tangent. Round your answer to three decimal places.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 30 GEOMETRY
Lesson 30: Trigonometry and the Pythagorean Theorem Date: 9/26/14 S.195
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7. The right triangle shown is taken from a slice of a right rectangular pyramid with a square base.
a. Find the height of the pyramid (to the nearest tenth).
b. Find the lengths of the sides of the base of the pyramid (to the nearest tenth).c. Find the lateral surface area of the right rectangular pyramid.
8. A machinist is fabricating a wedge in the shape of a right triangular prism. One acute angle of the right triangularbase is 33ยฐ, and the opposite side is 6.5 ๐๐. Find the length of the edges labeled ๐ and ๐ using sine, cosine, and/ortangent. Round your answer to the nearest thousandth of a centimeter.
9. Let sin ๐ = ๐๐
, where ๐, ๐ > 0. Express tan ๐ and cos ๐ in terms of ๐ and ๐.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area Date: 9/26/14 S.196
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Lesson 31: Using Trigonometry to Determine Area
Classwork
Opening Exercise
Three triangles are presented below. Determine the areas for each triangle, if possible. If it is not possible to find the area with the provided information, describe what is needed in order to determine the area.
Is there a way to find the missing information?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area Date: 9/26/14 S.197
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Example 1
What if the third side length of the triangle were provided? Is it possible to determine the area of the triangle now? Find the area of โณ ๐บ๐ป๐ผ.
Example 2
A farmer is planning how to divide his land for planting next yearโs crops. A triangular plot of land is left with two known side lengths measuring 500 m and 1,700 m.
What could the farmer do next in order to find the area of the plot?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area Date: 9/26/14 S.198
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Exercises 1
A real estate developer and her surveyor are searching for their next piece of land to build on. They each examine a plot of land in the shape of โณ ๐ด๐ต๐ถ. The real estate developer measures the length of ๐ด๐ต and ๐ด๐ถ and finds them to both be approximately 4,000 feet, and the included angle has a measure of approximately 50ห. The surveyor measures the length of ๐ด๐ถ and ๐ต๐ถ and finds the lengths to be approximately 4,000 feet and 3,400 feet, respectively, and measures the angle between the two sides to be approximately 65ยฐ.
a. Draw a diagram that models the situation, labeling all lengths and angle measures.
b. The real estate developer and surveyor each calculate the area of the plot of land and both find roughly thesame area. Show how each person calculated the area; round to the nearest hundred. Redraw the diagramwith only the relevant labels for both the real estate agent and surveyor.
c. What could possibly explain the difference between the real estate agent and surveyorโs calculated areas?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area Date: 9/26/14 S.199
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Problem Set
Find the area of each triangle. Round each answer to the nearest tenth.
1.
2.
3.
4.
5. In โณ ๐ท๐ธ๐น, ๐ธ๐น = 15, ๐ท๐น = 20, and โ ๐น = 63. Determine the area of the triangle. Round to the nearest tenth.
6. A landscape designer is designing a flower garden for a triangular area that is bounded on two sides by the clientโshouse and driveway. The length of the edges of the garden along the house and driveway are 18 ft. and 8 ft.respectively, and the edges come together at an angle of 80ยฐ. Draw a diagram, and then find the area of the gardento the nearest square foot.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area Date: 9/26/14 S.200
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7. A right rectangular pyramid has a square base with sides of length 5. Each lateral face of the pyramid is an isoscelestriangle. The angle on each lateral face between the base of the triangle and the adjacent edge is 75ยฐ. Find thesurface area of the pyramid to the nearest tenth.
8. The Pentagon Building in Washington D.C. is built in the shape of a regular pentagon. Each side of the pentagonmeasures 921 ft. in length. The building has a pentagonal courtyard with the same center. Each wall of the centercourtyard has a length of 356 ft. What is the approximate area of the roof of the Pentagon Building?
9. A regular hexagon is inscribed in a circle with a radius of 7. Find the perimeter and area of the hexagon.
10. In the figure below, โ ๐ด๐ธ๐ต is acute. Show that Area(โณ ๐ด๐ต๐ถ) = 12๐ด๐ถ โ ๐ต๐ธ โ sinโ ๐ด๐ธ๐ต.
11. Let ๐ด๐ต๐ถ๐ท be a quadrilateral. Let ๐ค be the measure of the acute angle formed by diagonals ๐ด๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ต๐ท๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ. Show thatArea(๐ด๐ต๐ถ๐ท) = 1
2๐ด๐ถ โ ๐ต๐ท โ sin๐ค.
(Hint: Apply the result from Problem 10 to โณ ๐ด๐ต๐ถ and โณ ๐ด๐ถ๐ท.)
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 32 GEOMETRY
Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle Date: 9/26/14 S.201
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Lesson 32: Using Trigonometry to Find Side Lengths of an Acute
Triangle
Classwork
Opening Exercise
a. Find the lengths of ๐ and ๐.
b. Find the lengths of ๐ฅ and ๐ฆ. How is this different from part (a)?
Example 1
A surveyor needs to determine the distance between two points ๐ด and ๐ต that lie on opposite banks of a river. A point ๐ถ is chosen 160 meters from point ๐ด, on the same side of the river as ๐ด. The measures of angles โ ๐ต๐ด๐ถ and โ ๐ด๐ถ๐ต are 41ห and 55ห, respectively. Approximate the distance from ๐ด to ๐ต to the nearest meter.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 32 GEOMETRY
Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle Date: 9/26/14 S.202
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Exercises 1โ2
1. In โณ ๐ด๐ต๐ถ, ๐โ ๐ด = 30, ๐ = 12, and ๐ = 10. Find sinโ ๐ต. Include a diagram in your answer.
2. A car is moving towards a tunnel carved out of the base of a hill. As the accompanying diagram shows, the top ofthe hill, ๐ป, is sighted from two locations, ๐ด and ๐ต. The distance between ๐ด and ๐ต is 250 ft. What is the height, โ, ofthe hill to the nearest foot?
Example 2
Our friend the surveyor from Example 1 is doing some further work. He has already found the distance between points ๐ด and ๐ต (from Example 1). Now he wants to locate a point ๐ท that is equidistant from both ๐ด and ๐ต and on the same side of the river as ๐ด. He has his assistant mark the point ๐ท so that the angles โ ๐ด๐ต๐ท and โ ๐ต๐ด๐ท both measure 75ห. What is the distance between ๐ท and ๐ด to the nearest meter?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 32 GEOMETRY
Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle Date: 9/26/14 S.203
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Exercise 3
3. Parallelogram ๐ด๐ต๐ถ๐ท has sides of lengths 44 mm and 26 mm, and one of the angles has a measure of 100ห.Approximate the length of diagonal ๐ด๐ถ to the nearest mm.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 32 GEOMETRY
Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle Date: 9/26/14 S.204
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Problem Set
1. Given โณ ๐ด๐ต๐ถ, ๐ด๐ต = 14, โ ๐ด = 57.2ยฐ, and โ ๐ถ = 78.4ยฐ, calculate themeasure of angle ๐ต to the nearest tenth of a degree, and use theLaw of Sines to find the lengths of ๐ด๐ถ and ๐ต๐ถ to the nearest tenth.
Calculate the area of โณ ๐ด๐ต๐ถ to the nearest square unit.
2. Given โณ ๐ท๐ธ๐น, โ ๐น = 39ยฐ, and ๐ธ๐น = 13, calculate the measure of โ ๐ธ, and usethe Law of Sines to find the lengths of ๐ท๐น๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ท๐ธ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to the nearest hundredth.
3. Does the law of sines apply to a right triangle? Based on โณ ๐ด๐ต๐ถ, the following ratios were set up according to thelaw of sines.
sinโ ๐ด๐
=sinโ ๐ต๐
=sin 90๐
Fill in the partially completed work below.
sinโ ๐ด๐
=sin 90๐
sinโ ๐ด๐
=๐
sinโ ๐ด =๐
sinโ ๐ต๐
=sin 90๐
sinโ ๐ต๐
=๐
sinโ ๐ต =๐
What conclusions can we draw?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 32 GEOMETRY
Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle Date: 9/26/14 S.205
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4. Given quadrilateral ๐บ๐ป๐พ๐ฝ, โ ๐ป = 50ยฐ, โ ๐ป๐พ๐บ = 80ยฐ, โ ๐พ๐บ๐ฝ = 50ยฐ, โ ๐ฝ is a right angle and ๐บ๐ป = 9 in., use the law ofsines to find the length of ๐บ๐พ, and then find the lengths of ๐บ๐ฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ฝ๐พ๏ฟฝ๏ฟฝ๏ฟฝ to the nearest tenth of an inch.
5. Given triangle ๐ฟ๐๐, ๐ฟ๐ = 10, ๐ฟ๐ = 15, and โ ๐ฟ = 38ยฐ, use the law of cosines to find the length of ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ to thenearest tenth.
6. Given triangle ๐ด๐ต๐ถ, ๐ด๐ถ = 6, ๐ด๐ต = 8, and โ ๐ด = 78ยฐ. Draw a diagram of triangle ๐ด๐ต๐ถ, and use the law of cosines tofind the length of ๐ต๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
Calculate the area of triangle ๐ด๐ต๐ถ.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying Laws of Sines and Cosines Date: 9/26/14 S.206
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Lesson 33: Applying Laws of Sines and Cosines
Classwork
Opening Exercise
For each triangle shown below, identify the method (Pythagorean theorem, law of sines, law of cosines) you would use to find each length ๐ฅ.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying Laws of Sines and Cosines Date: 9/26/14 S.207
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying Laws of Sines and Cosines Date: 9/26/14 S.208
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Example 1
Find the missing side length in โณ ๐ด๐ต๐ถ.
Example 2
Find the missing side length in โณ ๐ด๐ต๐ถ.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying Laws of Sines and Cosines Date: 9/26/14 S.209
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Exercises 1โ6
Use the Laws of sines and cosines to find all missing side lengths for each of the triangles in the Exercises below. Round your answers to the tenths place.
1. Use the triangle below to complete this exercise.
a. Identify the method (Pythagorean theorem, law of sines, law of cosines)you would use to find each of the missing lengths of the triangle.Explain why the other methods cannot be used.
b. Find the lengths of ๐ด๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ด๐ต๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ.
2. Your school is challenging classes to compete in a triathlon. The race begins with a swim along the shore, thencontinues with a bike ride for 4 miles. School officials want the race to end at the place it began so after the 4-milebike ride racers must turn 30ห and run 3.5 mi. directly back to the starting point.
a. Identify the method (Pythagorean theorem, law of sines, law ofcosines) you would use to find the total length of the race.Explain why the other methods cannot be used.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying Laws of Sines and Cosines Date: 9/26/14 S.210
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b. Determine the total length of the race. Round your answer to the tenths place.
3. Two lighthouses are 30 mi. apart on each side of shorelines running north and south, as shown. Each lighthousekeeper spots a boat in the distance. One lighthouse keeper notes the location of the boat as 40ห east of south, andthe other lighthouse keeper marks the boat as 32ห west of south. What is the distance from the boat to each of thelighthouses at the time it was spotted? Round your answers to the nearest mile.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying Laws of Sines and Cosines Date: 9/26/14 S.211
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4. A pendulum 18 in. in length swings 72ห from right to left. What is the difference between the highest and lowestpoint of the pendulum? Round your answer to the hundredths place, and explain how you found it.
5. What appears to be the minimum amount of information about a triangle that must be given in order to use the lawof sines to find an unknown length?
6. What appears to be the minimum amount of information about a triangle that must be given in order to use the lawof cosines to find an unknown length?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying Laws of Sines and Cosines Date: 9/26/14 S.212
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Problem Set
1. Given triangle ๐ธ๐น๐บ, ๐น๐บ = 15, angle ๐ธ has measure of 38ยฐ, and angle ๐น has measure 72ยฐ, find the measures of theremaining sides and angle to the nearest tenth. Justify your method.
2. Given triangle ๐ด๐ต๐ถ, angle ๐ด has measure of 75ยฐ, ๐ด๐ถ = 15.2, and ๐ด๐ต = 24, find ๐ต๐ถ to the nearest tenth. Justifyyour method.
3. James flies his plane from point ๐ด at a bearing of 32ยฐ east of north, averaging speed of 143 miles per hour for 3hours, to get to an airfield at point ๐ต. He next flies 69ยฐ west of north at an average speed of 129 miles per hour for4.5 hours to a different airfield at point ๐ถ.
a. Find the distance from ๐ด to ๐ต.
b. Find the distance from ๐ต to ๐ถ,c. Find the measure of angle ๐ด๐ต๐ถ.
d. Find the distance from ๐ถ to ๐ด.
e. What length of time can James expect the return tripfrom ๐ถ to ๐ด to take?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 33 GEOMETRY
Lesson 33: Applying Laws of Sines and Cosines Date: 9/26/14 S.213
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4. Mark is deciding on the best way to get from point ๐ด to point ๐ต as shown on the map of Crooked Creek to go fishing.He sees that if he stays on the north side of the creek, he would have to walk around a triangular piece of privateproperty (bounded by ๐ด๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ and ๐ต๐ถ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ). His other option is to cross the creek at ๐ด and take a straight path to ๐ต, whichhe knows to be a distance of 1.2 mi. The second option requires crossing the water, which is too deep for his bootsand very cold. Find the difference in distances to help Mark decide which path is his better choice.
5. If you are given triangle ๐๐๐, and the measures of two of its angles, and two of its sides, would it be appropriate toapply the law of sines or the law of cosines to find the remaining side? Explain.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 34 GEOMETRY
Lesson 34: Unknown Angles Date: 9/26/14 S.214
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Lesson 34: Unknown Angles
Classwork
Opening Exercise
a. Dan was walking through a forest when he came upon a sizable tree. Dan estimated he was about 40 metersaway from a tree when he measured the angle of elevation between the horizontal and the top of the tree tobe 35 degrees. If Dan is about 2 meters tall, about how tall is the tree?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 34 GEOMETRY
Lesson 34: Unknown Angles Date: 9/26/14 S.215
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b. Dan was pretty impressed with this tree โฆ until he turned around and saw a bigger one, also 40 meters awaybut in the other direction. โWow,โ he said. โI bet that tree is at least 50 meters tall!โ Then he thought amoment. โHmm โฆ if it is 50 meters tall, I wonder what angle of elevation I would measure from my eye levelto the top of the tree?โ What angle will Dan find if the tree is 50 meters tall? Explain your reasoning.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 34 GEOMETRY
Lesson 34: Unknown Angles Date: 9/26/14 S.216
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Exercises 1โ5
1. Find the measure of angles aโd to the nearest degree.a.
b.
c.
d.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 34 GEOMETRY
Lesson 34: Unknown Angles Date: 9/26/14 S.217
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2. Shelves are being built in a classroom to hold textbooks and other supplies.The shelves will extend 10 in from the wall. Support braces will need to beinstalled to secure the shelves. The braces will be attached to the end of theshelf and secured 6 in below the shelf on the wall. What angle measure willthe brace and the shelf make?
3. A 16 ft ladder leans against a wall. The foot of the ladder is 7 ft from the wall.
a. Find the vertical distance from the ground to the point where the top of the ladder touches the wall.
b. Determine the measure of the angle formed by the ladder and the ground.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 34 GEOMETRY
Lesson 34: Unknown Angles Date: 9/26/14 S.218
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4. A group of friends have hiked to the top of the Mile High Mountain. When they look down, they can see theircampsite, which they know is approximately 3 miles from the base of the mountain.
a. Sketch a drawing of the situation.
b. What is the angle of depression?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 34 GEOMETRY
Lesson 34: Unknown Angles Date: 9/26/14 S.219
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5. A roller coaster travels 80 ft of track from the loading zone before reaching its peak. The horizontal distancebetween the loading zone and the base of the peak is 50 ft.
a. Model the situation using a right triangle.
b. At what angle is the roller coaster rising according to the model?
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 34 GEOMETRY
Lesson 34: Unknown Angles Date: 9/26/14 S.220
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Problem Set
1. For each triangle shown, use the given information to find the indicated angle to the nearest degree.
a.
b.
Lesson Summary
In the same way that mathematicians have named certain ratios within right triangles, they have also developed terminology for identifying angles in a right triangle, given the ratio of the sides. Mathematicians will often use the prefix โarcโ to define these because an angle is not just measured as an angle, but also as a length of an arc on the unit circle.
Given a right triangle โณ ๐ด๐ต๐ถ, the measure of angle ๐ถ can be found in the following ways:
We can write similar statements to determine the measure of angle ๐ด.
We can use a calculator to help us determine the values of arcsin, arccos, and arctan. Most calculators show these buttons as โsin-1,โ โcos-1,โ and โtan-1.โ This subject will be addressed again in future courses.
arcsin ๏ฟฝ๐ด๐ต๐ด๐ถ๏ฟฝ = ๐โ ๐ถ
arccos ๏ฟฝ๐ต๐ถ๐ด๐ถ๏ฟฝ = ๐โ ๐ถ
arctan ๏ฟฝ๐ด๐ต๐ต๐ถ๏ฟฝ = ๐โ ๐ถ
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 34 GEOMETRY
Lesson 34: Unknown Angles Date: 9/26/14 S.221
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c.
2. Solving a right triangle means using given information to find all the angles and side lengths of the triangle. Use๐๐๐๐ ๐๐ and ๐๐๐๐๐๐ , along with the given information, to solve right triangle ๐ด๐ต๐ถ if leg ๐ด๐ถ = 12 and hypotenuse๐ด๐ต = 15.
Once you have found the measure of one of the acute angles in the right triangle, can you find the measure of theother acute angle using a different method than those used in this lesson? Explain.
3. A pendulum consists of a spherical weight suspended at the endof a string whose other end is anchored at a pivot point ๐. Thedistance from ๐ to the center of the pendulumโs sphere, ๐ต, is 6inches. The weight is held so that the string is taught andhorizontal, as shown to the right, and then dropped.a. What type of path does the pendulumโs weight take as it
swings?
b. Danni thinks that for every vertical drop of 1 inch that thependulumโs weight makes, the degree of rotation is 15ยฐ. Doyou agree or disagree with Danni? As part of yourexplanation, calculate the degree of rotation for everyvertical drop of 1 inch from 1 to 6 inches.
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NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 34 GEOMETRY
Lesson 34: Unknown Angles Date: 9/26/14 S.222
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4. A stone tower was built on unstable ground and the soil beneath itsettled under its weight causing the tower to lean. The cylindricaltower has a diameter of 17 meters. The height of the tower on thelow side measured 46.3 meters and on the high side measured 47.1meters. To the nearest tenth of a degree, find the angle that thetower has leaned from its original vertical position.
5. Doug is installing a surveillance camera inside a convenience store. He mounts the camera 8 ft above the groundand 15 ft horizontally from the storeโs entrance. The camera is being installed to monitor every customer thatenters and exits the store. At what angle of depression should Doug set the camera to capture the faces of allcustomers?