Post on 05-Jan-2016
Holt Geometry
7-4 Applying Properties of Similar Triangles
Warm UpSolve each proportion.
1. 2.
3.
AB = 16 QR = 10.5
x = 21
Holt Geometry
7-4 Applying Properties of Similar Triangles7-4 Applying Properties of Similar Triangles
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Geometry
7-4 Applying Properties of Similar Triangles
Use properties of similar triangles to find segment lengths.Apply proportionality and triangle angle bisector theorems.
Objectives
Holt Geometry
7-4 Applying Properties of Similar Triangles
Artists use mathematical techniques to make two-dimensional paintings appear three-dimensional. The invention of perspective was based on the observation that far away objects look smaller andcloser objects look larger.
Mathematical theorems like the Triangle Proportionality Theorem are important in making perspective drawings.
Holt Geometry
7-4 Applying Properties of Similar Triangles
Holt Geometry
7-4 Applying Properties of Similar Triangles
Example 1: Finding the Length of a Segment
Find US.
Substitute 14 for RU, 4 for VT, and 10 for RV.
Cross Products Prop.US(10) = 56
Divide both sides by 10.
It is given that , so by
the Triangle Proportionality Theorem.
Holt Geometry
7-4 Applying Properties of Similar Triangles
Check It Out! Example 1
Find PN.
Substitute in the given values.
Cross Products Prop.2PN = 15
PN = 7.5 Divide both sides by 2.
Use the Triangle Proportionality Theorem.
Holt Geometry
7-4 Applying Properties of Similar Triangles
Holt Geometry
7-4 Applying Properties of Similar Triangles
Example 2: Verifying Segments are Parallel
Verify that .
Since , by the Converse of the
Triangle Proportionality Theorem.
Holt Geometry
7-4 Applying Properties of Similar Triangles
Check It Out! Example 2
AC = 36 cm, and BC = 27 cm.
Verify that .
Since , by the Converse of the
Triangle Proportionality Theorem.
Holt Geometry
7-4 Applying Properties of Similar Triangles
Holt Geometry
7-4 Applying Properties of Similar Triangles
The previous theorems and corollary lead to the following conclusion.
Holt Geometry
7-4 Applying Properties of Similar Triangles
Example 4: Using the Triangle Angle Bisector Theorem
Find PS and SR.
Substitute the given values.
Cross Products Property
Distributive Property
by the ∆ Bisector Theorem.
40(x – 2) = 32(x + 5)
40x – 80 = 32x + 160
Holt Geometry
7-4 Applying Properties of Similar Triangles
Example 4 Continued
Simplify.
Divide both sides by 8.
Substitute 30 for x.
40x – 80 = 32x + 160
8x = 240
x = 30
PS = x – 2 SR = x + 5
= 30 – 2 = 28 = 30 + 5 = 35
Holt Geometry
7-4 Applying Properties of Similar Triangles
Check It Out! Example 4
Find AC and DC.
Substitute in given values.
Cross Products Theorem
So DC = 9 and AC = 16.
Simplify.
by the ∆ Bisector Theorem.
4y = 4.5y – 9
–0.5y = –9
Divide both sides by –0.5. y = 18
Holt Geometry
7-4 Applying Properties of Similar Triangles
Holt Geometry
7-4 Applying Properties of Similar Triangles
Find the length of each segment.
1. 2.
Lesson Quiz: Part I
SR = 25, ST = 15
Holt Geometry
7-4 Applying Properties of Similar Triangles
Lesson Quiz: Part II
3. Verify that BE and CD are parallel.
Since , by the
Converse of the ∆ Proportionality Thm.
7-5 Using Proportional Relationships7-5 Using Proportional Relationships
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
7-5 Using Proportional Relationships
indirect measurementscale drawingscale
Vocabulary
7-5 Using Proportional Relationships
Indirect measurement is any method that uses formulas, similar figures, and/or proportions to measure an object. The following example shows one indirect measurement technique.
7-5 Using Proportional Relationships
Whenever dimensions are given in both feet and inches, you must convert them to either feet or inches before doing any calculations.
Helpful Hint
7-5 Using Proportional Relationships
Example 1: Measurement Application
Tyler wants to find the height of a telephone pole. He measured the pole’s shadow and his own shadow and then made a diagram. What is the height h of the pole?
7-5 Using Proportional Relationships
Example 1 Continued
Step 1 Convert the measurements to inches.
Step 2 Find similar triangles.
Because the sun’s rays are parallel, A F. Therefore ∆ABC ~ ∆FGH by AA ~.
AB = 7 ft 8 in. = (7 12) in. + 8 in. = 92 in.
BC = 5 ft 9 in. = (5 12) in. + 9 in. = 69 in.
FG = 38 ft 4 in. = (38 12) in. + 4 in. = 460 in.
7-5 Using Proportional Relationships
Example 1 Continued
Step 3 Find h.
The height h of the pole is 345 inches, or 28 feet 9 inches.
Corr. sides are proportional.
Substitute 69 for BC, h for GH, 92 for AB, and 460 for FG.
Cross Products Prop.92h = 69 460
Divide both sides by 92.h = 345
7-5 Using Proportional Relationships
Check It Out! Example 1
A student who is 5 ft 6 in. tall measured shadows to find the height LM of a flagpole. What is LM?
Step 1 Convert the measurements to inches.
GH = 5 ft 6 in. = (5 12) in. + 6 in. = 66 in.
JH = 5 ft = (5 12) in. = 60 in.
NM = 14 ft 2 in. = (14 12) in. + 2 in. = 170 in.
7-5 Using Proportional Relationships
Check It Out! Example 1 Continued
Step 2 Find similar triangles.
Because the sun’s rays are parallel, L G. Therefore ∆JGH ~ ∆NLM by AA ~.
Step 3 Find h.
Corr. sides are proportional.
Substitute 66 for BC, h for LM, 60 for JH, and 170 for MN.
Cross Products Prop.
Divide both sides by 60.
60(h) = 66 170
h = 187
The height of the flagpole is 187 in., or 15 ft. 7 in.
7-5 Using Proportional Relationships
A scale drawing represents an object as smaller than or larger than its actual size. The drawing’s scale is the ratio of any length in the drawingto the corresponding actual length. For example, on a map with a scale of 1 cm : 1500 m, one centimeter on the map represents 1500 m in actual distance.
7-5 Using Proportional Relationships
A proportion may compare measurements that have different units.
Remember!
7-5 Using Proportional Relationships
Check It Out! Example 3
The rectangular central chamber of the Lincoln Memorial is 74 ft long and 60 ft wide. Make a scale drawing of the floor of the chamber using a scale of 1 in.:20 ft.
7-5 Using Proportional Relationships
Set up proportions to find the length l and width w of the scale drawing.
20w = 60
w = 3 in
Check It Out! Example 3 Continued
3.7 in.
3 in.
7-5 Using Proportional Relationships
7-5 Using Proportional Relationships
7-5 Using Proportional Relationships
Example 4: Using Ratios to Find Perimeters and Areas
Given that ∆LMN:∆QRT, find the perimeter P and area A of ∆QRS.
The similarity ratio of ∆LMN to
∆QRS is
By the Proportional Perimeters and Areas Theorem,
the ratio of the triangles’ perimeters is also , and
the ratio of the triangles’ areas is
7-5 Using Proportional Relationships
Example 4 Continued
Perimeter Area
The perimeter of ∆QRS is 25.2 cm, and the area is 29.4 cm2.
13P = 36(9.1)
P = 25.2
132A = (9.1)2(60)
A = 29.4 cm2
7-5 Using Proportional Relationships
Check It Out! Example 4
∆ABC ~ ∆DEF, BC = 4 mm, and EF = 12 mm. If P = 42 mm and A = 96 mm2 for ∆DEF, find the perimeter and area of ∆ABC.
Perimeter Area
The perimeter of ∆ABC is 14 mm, and the area is 10.7 mm2.
12P = 42(4)
P = 14 mm
122A = (4)2(96)
7-5 Using Proportional Relationships
7-5 Using Proportional Relationships
Lesson Quiz: Part I
1. Maria is 4 ft 2 in. tall. To find the height of a flagpole, she measured her shadow and the pole’s shadow. What is the height h of the flagpole?
2. A blueprint for Latisha’s bedroom uses a scale of 1 in.:4 ft. Her bedroom on the blueprint is 3 in. long. How long is the actual room?
25 ft
12 ft
7-5 Using Proportional Relationships
Lesson Quiz: Part II
3. ∆ABC ~ ∆DEF. Find the perimeter and area of ∆ABC.
P = 27 in., A = 31.5 in2
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane7-6 Dilations and Similarity
in the Coordinate Plane
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Warm UpSimplify each radical.
1. 2. 3.
Find the distance between each pair of points. Write your answer in simplest radical form.
4. C (1, 6) and D (–2, 0)
5. E(–7, –1) and F(–1, –5)
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Apply similarity properties in the coordinate plane.
Use coordinate proof to prove figures similar.
Objectives
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
dilationscale factor
Vocabulary
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
If the scale factor of a dilation is greater than 1 (k > 1), it is an enlargement. If the scale factor is less than 1 (k < 1), it is a reduction.
Helpful Hint
Holt Geometry
7-6 Dilations and Similarity in the Coordinate PlaneExample 1: Computer Graphics Application
Draw the border of the photo after a
dilation with scale factor
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Example 1 Continued
RectangleABCD
Step 1 Multiply the vertices of the photo A(0, 0), B(0,
4), C(3, 4), and D(3, 0) by
RectangleA’B’C’D’
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Example 1 Continued
Step 2 Plot points A’(0, 0), B’(0, 10), C’(7.5, 10), and D’(7.5, 0).
Draw the rectangle.
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Check It Out! Example 1
What if…? Draw the border of the original photo
after a dilation with scale factor
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Check It Out! Example 1 Continued Step 1 Multiply the vertices of the photo A(0, 0), B(0,
4), C(3, 4), and D(3, 0) by
RectangleABCD
RectangleA’B’C’D’
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Check It Out! Example 1 Continued
Step 2 Plot points A’(0, 0), B’(0, 2), C’(1.5, 2), and D’(1.5, 0).
Draw the rectangle.
A’ D’
B’ C’
01.5
2
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Example 2: Finding Coordinates of Similar Triangle
Given that ∆TUO ~ ∆RSO, find the coordinates of U and the scale factor.
Since ∆TUO ~ ∆RSO,
Substitute 12 for RO, 9 for TO, and 16 for OY.
12OU = 144 Cross Products Prop.
OU = 12 Divide both sides by 12.
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Example 2 Continued
U lies on the y-axis, so its x-coordinate is 0. Since OU = 12, its y-coordinate must be 12. The coordinates of U are (0, 12).
So the scale factor is
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Example 3: Proving Triangles Are Similar
Prove: ∆EHJ ~ ∆EFG.
Given: E(–2, –6), F(–3, –2), G(2, –2), H(–4, 2), and J(6, 2).
Step 1 Plot the points and draw the triangles.
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Example 3 Continued
Step 2 Use the Distance Formula to find the side lengths.
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Example 3 Continued
Step 3 Find the similarity ratio.
= 2 = 2
Since and E E, by the Reflexive Property,
∆EHJ ~ ∆EFG by SAS ~ .
Holt Geometry
7-6 Dilations and Similarity in the Coordinate PlaneExample 4: Using the SSS Similarity Theorem
Verify that ∆A'B'C' ~ ∆ABC.
Graph the image of ∆ABC
after a dilation with scale
factor
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Example 4 Continued
Step 1 Multiply each coordinate by to find the
coordinates of the vertices of ∆A’B’C’.
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Step 2 Graph ∆A’B’C’.
Example 4 Continued
C’ (4, 0)
A’ (0, 2)
B’ (2, 4)
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Step 3 Use the Distance Formula to find the side lengths.
Example 4 Continued
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Example 4 Continued
Step 4 Find the similarity ratio.
Since , ∆ABC ~ ∆A’B’C’ by SSS ~.
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Check It Out! Example 4
Graph the image of ∆MNP after a dilation with scale factor 3. Verify that ∆M'N'P' ~ ∆MNP.
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Check It Out! Example 4 Continued
Step 1 Multiply each coordinate by 3 to find the coordinates of the vertices of ∆M’N’P’.
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Step 2 Graph ∆M’N’P’.
Check It Out! Example 4 Continued
-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
-7-6-5-4-3-2-1
1234567
X
Y
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Step 3 Use the Distance Formula to find the side lengths.
Check It Out! Example 4 Continued
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Check It Out! Example 4 Continued
Step 4 Find the similarity ratio.
Since , ∆MNP ~ ∆M’N’P’ by SSS ~.
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Lesson Quiz: Part I
1. Given X(0, 2), Y(–2, 2), and Z(–2, 0), find the
coordinates of X', Y, and Z' after a dilation with
scale factor –4.
2. ∆JOK ~ ∆LOM. Find the coordinates of M and the scale factor.
X'(0, –8); Y'(8, –8); Z'(8, 0)
Holt Geometry
7-6 Dilations and Similarity in the Coordinate Plane
Lesson Quiz: Part II
3. Given: A(–1, 0), B(–4, 5), C(2, 2), D(2, –1),
E(–4, 9), and F(8, 3)
Prove: ∆ABC ~ ∆DEF
Therefore, and ∆ABC ~ ∆DEF
by SSS ~.