1 Standard: G.SRT.1 Objective: I can dilate an image when the center of dilation is the origin. A dilation is a transformation that produces an image that is the same shape as the original figure but the image is different size. The dilation uses a center and a scale factor to create a proportional figure. The center of dilation is a fixed point in the plane about which all points are expanded or contracted. The scale factor is the ratio of the new image to the original. There are two ways to dilate a figure when the origin is the center of dilation. One method is to multiply the x and the y in each vertex by the scale factor. Example:

a) Center (0,0); Dilate by a scale factor of 2

b) Center (0,0); Dilate by a scale factor of 1

3

The other way is to use the slope. This method will help better prepare you for tomorrow, so I suggest using this method.

a) Center (0,0); Dilate by a scale factor of 2

b) Center (0,0); Dilate by a scale factor of 1

3

Practice A: Dilate each image using the given scale factor. The center of dilation for each is the origin.

1. Scale Factor: 2

2. Scale Factor: 1/2

2

2. Scale Factor: 1/3

4. Scale Factor: 3

5. Scale Factor: 2

6. Scale Factor: 1/3

Standard: G.SRT.1 Objective: I can dilate an image when the center of dilation is not the origin. Not all images are dilated around the origin. Here are two examples of images that were not dilated about the origin.

Two figures are similar if and only if there is a dilation that maps one figure onto the other. In the new figure, corresponding angles are congruent and corresponding sides are proportional to the original figure. You can denote that two figures are similar by using the symbol ~. For example. ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹.

3 Examples:

a) Center: (2,2); scale factor: 2

b) Center: (5,8); scale factor: ½

Practice B Dilate each image.

Center (0, 0); scale factor ½

Center (0, 0); scale factor 3

Center (-3, -3); scale factor 2

Center (-5, 5); scale factor 4

Center (2, -3); scale factor ½

Center (-1, 0); scale factor 2

Standard: G.STR.2 Objective: I can use theorems about triangles to help solve for missing sides and angles. Definitions:

- The angles inside of a triangle are called the interior angles.

- A triangle with at least two congruent sides is called an isosceles triangle. In an isosceles triangle, the angles that are opposite the congruent sides are called the base angles.

- A point that is halfway between the endpoints of a segment is called the midpoint. Point 𝐶 is the midpoint of 𝐴𝐵̅̅ ̅̅ .

4 - A segment whose endpoints are the midpoints of two sides of a triangle is called the

midsegment of a triangle.

- The line connecting midpoints to the opposite vector of a triangle is called the median. Point 𝑆 is the midpoint of 𝐴𝐶̅̅ ̅̅ . Point 𝑄 is the midpoint of 𝐵𝐶̅̅ ̅̅ . Point 𝑅 is the midpoint of

𝐴𝐵̅̅ ̅̅ . - The point where all three medians of a triangle intersect is called a centroid. Point 𝑇 is

the centroid of ∆𝐴𝐵𝐶. Theorems: Angle Sum Theorem: The sum of the measures of the angles of a triangle is 180°. Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Triangle Midsegment Theorem: A midsegment of a triangle is parallel to one side of the triangle, and its length is one-half the length of that side

Triangle Proportionality Theorem: If a line is parallel to one side of a triangle, then it divides the other two sides

proportionally.

Once you have found that two triangles are similar, you can use the theorems above to find missing sides and angles. Examples:

a) Find the missing angles.

b) Find the missing angles.

c) Find 𝑀𝐿̅̅ ̅̅ and 𝐿𝐾̅̅ ̅̅

d) Find 𝑀𝑁̅̅ ̅̅ ̅ and 𝑀𝐽̅̅ ̅̅

5 e) Find 𝐾𝑀̅̅ ̅̅ ̅ and 𝑂𝑃̅̅ ̅̅

f) Find 𝐽𝐾̅̅ ̅ and 𝐽�̅�

Practice C Find the missing angles

1.

2.

3.

4.

Find the missing sides

5.

6.

7.

8.

6

9.

10.

11. Find 𝑥, 𝑄𝑇, and 𝑇𝑅 if 𝑄𝑇 = 𝑥 + 6, 𝑆𝑅 =12, 𝑃𝑆 = 27, and 𝑇𝑅 = 𝑥 − 4.

12. Find 𝐵𝐶, 𝐹𝐸, 𝐶𝐷, and 𝐷𝐸 if 𝐴𝐵 = 6, 𝐴𝐹 = 8, 𝐵𝐶 = 𝑥, 𝐶𝐷 = 𝑦, 𝐷𝐸 = 2𝑦 − 3, and 𝐹𝐸 =

𝑥 +10

3.

Standard: G.STR.2 Objective: I can use theorems about parallelograms to find missing sides and angles. Vocabulary:

- A polygon with four sides is called a quadrilateral.

- A quadrilateral with two pairs of paralles sides is called a parallelogram.

- A segment that connects two nonconsecutive vertices is called a diagonal.

- A parallelogram with four right angles is called a rectangle. Theorems:

- If a quadrilateral is a parallelogram, then its opposite sides are congruent.

- If a quadrilateral is a parallelogram, then its opposite angles are congruaent.

7 - If a quadrilateral is a parallelogram, then its diagonals bisect each other.

- If a diagnonal of a quadrilatoral bisect eachother, the the quadrilateral is a parallelogram.

- If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

If you know a polygon is a parallelogram, then you can use the previous theorems to find missing angles and side lenghts. Examples: Find the values of the variables in each parallelogram.

a)

b)

c)

d)

e)

f)

8

Practice D Complete the statement about ⊡ 𝐷𝐸𝐹𝐺

1. 𝐷𝐺̅̅ ̅̅ ∥ 2. 𝐷𝐸̅̅ ̅̅ ∥ 3. 𝐺𝐻̅̅ ̅̅ ≅ 4. ∠𝐷𝐸𝐹 ≅ 5. ∠𝐺𝐷𝐸 ≅ 6. ∠𝐷𝐻𝐸 ≅ Find the x and y in each paralelogram.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

9 Standard: G.STR.3 Objective: I can use and understand Coordinate Geometry

Find the coordinate of 𝐶 such that 𝐴𝐶

𝐶𝐵=

2

3

Find the length of 𝐴𝐵 using the distance formula.

Using the segmented addition postulate 𝐴𝐶 + 𝐶𝐵 = 𝐴𝐵 and solve for 𝐴𝐶

Substitute 𝐴𝐶 = 𝐴𝐵 − 𝐶𝐵 into 𝐴𝐶

𝐶𝐵=

2

3.

Clear the denominators, then get like terms on the same side.

Substitute the distance you found in the first step in for 𝐴𝐵, and solve.

Substitute 𝐶𝐵 and 𝐴𝐶 into 𝐴𝐶 + 𝐶𝐵 = 𝐴𝐵 and solve the for the remaining one.

10 To find the coordinates of the point 𝐶, you will

need to draw a right triangle with 𝐴𝐵̅̅ ̅̅ as the

hypotenuse. Then draw a line from point 𝐶

perpendicular to each leg of the triangle. The

two small Triangles created within the large

triangle are similar to each other and the large

triangle by AA Similar. Therefore the ratio of

their sides is equal.

You need the coordinate of point F to find the x-coordinate of point 𝐶. Use the ratio to find 𝐴𝐹.

Point 𝐹 will have the same y-coordiante as point 𝐴 +𝑤ℎ𝑎𝑡 𝑦𝑜𝑢 𝑓𝑜𝑢𝑛𝑑 𝑎𝑏𝑜𝑣𝑒.

You need the coordinates of Point 𝐸 to find the y-coordinate of point 𝐶. Use the Ratio to find 𝐷𝐸

The Point E will have the same x-coordinate as point 𝐷. The y-coordinate will be the same as 𝐷 +𝑤ℎ𝑎𝑡 𝑦𝑜𝑢 𝑓𝑜𝑢𝑛𝑑 𝑎𝑏𝑜𝑣𝑒

Write as an ordered pair

11

Examples

1. 𝐶 is between 𝐴 and 𝐵 with 𝐴(−4,1) and 𝐵(−1, −3). Find the coordinates of 𝐶 such that 𝐴𝐶

𝐶𝐵=

2

3 .

2. 𝐶 is between 𝐴 and 𝐵 with 𝐴(−7, −4) and 𝐵(9,8). Find the coordinates of 𝐶 such that

𝐴𝐶

𝐶𝐵=

3

4 .

3. 𝐶 is between 𝐴 and 𝐵 with 𝐴(7, −2) and 𝐵(−5,3). Find the coordinates of 𝐶 such that

𝐴𝐶

𝐶𝐵=

1

4 .

12

Practice E Find point C for each of the following

1. 𝐶 is between 𝐴 and 𝐵 with 𝐴(3,2) and 𝐵(−3,2). Find the coordinates of 𝐶 such that 𝐴𝐶

𝐶𝐵=

5

2 .

2. 𝐶 is between 𝐴 and 𝐵 with 𝐴(7,5) and 𝐵(−4, −3). Find the coordinates of 𝐶 such that

𝐴𝐶

𝐶𝐵=

3

4 .

3. 𝐶 is between 𝐴 and 𝐵 with 𝐴(−5,5) and 𝐵(−5,3). Find the coordinates of 𝐶 such that

𝐴𝐶

𝐶𝐵=

1

3 .

4. 𝐶 is between 𝐴 and 𝐵 with 𝐴(1,5) and 𝐵(−1, −5). Find the coordinates of 𝐶 such that

𝐴𝐶

𝐶𝐵=

1

2 .

5. 𝐶 is between 𝐴 and 𝐵 with 𝐴(−4,6) and 𝐵(−5,3). Find the coordinates of 𝐶 such that

𝐴𝐶

𝐶𝐵=

2

3 .

13

6. 𝐶 is between 𝐴 and 𝐵 with 𝐴(6,7) and 𝐵(2,4). Find the coordinates of 𝐶 such that 𝐴𝐶

𝐶𝐵=

2

5 .

7. 𝐶 is between 𝐴 and 𝐵 with 𝐴(3, −3) and 𝐵(−2,8). Find the coordinates of 𝐶 such that

𝐴𝐶

𝐶𝐵=

1

5 .

8. 𝐶 is between 𝐴 and 𝐵 with 𝐴(9,8) and 𝐵(2, −7). Find the coordinates of 𝐶 such that

𝐴𝐶

𝐶𝐵=

3

5 .

9. 𝐶 is between 𝐴 and 𝐵 with 𝐴(7, −2) and 𝐵(−4,4). Find the coordinates of 𝐶 such that

𝐴𝐶

𝐶𝐵= 4 .

10. 𝐶 is between 𝐴 and 𝐵 with 𝐴(7,1) and 𝐵(−5,6). Find the coordinates of 𝐶 such that

𝐴𝐶

𝐶𝐵=

1

4 .

14

Practice F Rewrite each expression as a radical. Then simplify the radical.

1. 84

3

2. 𝑥3

4 3. 𝑎5

9

4. 363

2

5. 𝑘3

2 6. 2𝑥1

5

7. (−64𝑥)2

3

8. −8𝑥5

3 9. (2𝑥)1

5

Rewrite each expression as a rational exponent

1. √113

2. √4227 3. √(−10)83

4. √524

5. √𝑥76

6. √𝑟3

7. √𝑤57 8. √𝑘23

9. √𝑧25