1. Warm Up
Solve each proportion. 3.
AB = 16
QR = 10.5
x = 21

2. Applying Properties 7-4 of Similar Triangles Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry

3. Objectives
Use properties of similar triangles to find segment lengths.
Apply proportionality and triangle angle bisector theorems.

4. 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 and
closer objects look larger.
Mathematical theorems like the Triangle Proportionality Theorem are important in making perspective drawings.

6. Example 1: Finding the Length of a Segment
Find US.
It is given that , so by the Triangle Proportionality Theorem.
Substitute 14 for RU,
4 for VT, and 10 for RV.
US(10) = 56
Cross Products Prop.
Divide both sides by 10.

7. Check It Out! Example 1
Find PN.
Use the Triangle Proportionality Theorem.
Substitute in the given values.
Cross Products Prop.
2PN = 15
PN = 7.5
Divide both sides by 2.

9. Example 2: Verifying Segments are Parallel
Verify that
Since , by the Converse of the Triangle Proportionality Theorem.

10. Check It Out! Example 2
AC = 36 cm, and BC = 27 cm. Verify that
Since , by the Converse of the
Triangle Proportionality Theorem.

12. The previous theorems and corollary lead to the following conclusion.

13. Example 4: Using the Triangle Angle Bisector Theorem
Find PS and SR.
by the ∆  Bisector Theorem.
Substitute the given values.
40(x – 2) = 32(x + 5)
Cross Products Property
40x – 80 = 32x + 160
Distributive Property

14. Example 4 Continued
40x – 80 = 32x + 160
8x = 240
Simplify.
x = 30
Divide both sides by 8.
Substitute 30 for x.
PS = x – 2
SR = x + 5
= 30 – 2 = 28
= = 35

15. Check It Out! Example 4
Find AC and DC.
by the ∆  Bisector Theorem.
Substitute in given values.
4y = 4.5y – 9
Cross Products Theorem
–0.5y = –9
Simplify.
y = 18
Divide both sides by –0.5.
So DC = 9 and AC = 16.

17. Lesson Quiz: Part I
Find the length of each segment.
SR = 25, ST = 15

18. Lesson Quiz: Part II
3. Verify that BE and CD are parallel.
Since , by the
Converse of the ∆ Proportionality Thm.

19. Using Proportional Relationships
7-5
Using Proportional Relationships
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry

20. Vocabulary
indirect measurement
scale drawing
scale

21. 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.

22. Whenever dimensions are given in both feet and inches, you must convert them to either feet or inches before doing any calculations.
Helpful Hint

23. 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?

24. Example 1 Continued
Step 1 Convert the measurements to inches.
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.
Step 2 Find similar triangles.
Because the sun’s rays are parallel, A  F. Therefore ∆ABC ~ ∆FGH by AA ~.

25. Example 1 Continued
Step 3 Find h.
Corr. sides are proportional.
Substitute 69 for BC, h for GH, 92 for AB, and 460 for FG.
92h = 69  460
Cross Products Prop.
h = 345
Divide both sides by 92.
The height h of the pole is 345 inches, or 28 feet 9 inches.

26. 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.

27. 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.
60(h) = 66  170
h = 187
Divide both sides by 60.
The height of the flagpole is 187 in., or 15 ft. 7 in.

28. 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 drawing
to 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.

29. A proportion may compare measurements that have different units.
Remember!

30. 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.

31. Check It Out! Example 3 Continued
Set up proportions to find the length l and width w of the scale drawing.
20w = 60
w = 3 in
3.7 in.
3 in.

34. 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

35. Example 4 Continued
Perimeter Area
13P = 36(9.1)
132A = (9.1)2(60)
P = 25.2
A = 29.4 cm2
The perimeter of ∆QRS is 25.2 cm, and the area is 29.4 cm2.

36. 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
12P = 42(4)
122A = (4)2(96)
P = 14 mm
The perimeter of ∆ABC is 14 mm, and the area is 10.7 mm2.

38. 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

39. Lesson Quiz: Part II
3. ∆ABC ~ ∆DEF. Find the perimeter and area of ∆ABC.
P = 27 in., A = 31.5 in2

40. Dilations and Similarity in the Coordinate Plane 7-6
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry

41. Warm Up
Simplify each radical.
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)

42. Objectives Apply similarity properties in the coordinate plane.
Use coordinate proof to prove figures similar.

43. Vocabulary
dilation
scale factor

44. 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

45. Example 1: Computer Graphics Application
Draw the border of the photo after a dilation with scale factor

46. 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
Rectangle
ABCD
Rectangle
A’B’C’D’

47. 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.

48. Check It Out! Example 1
What if…? Draw the border of the original photo after a dilation with scale factor

49. 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
Rectangle
ABCD
Rectangle
A’B’C’D’

50. 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’
1.5 2

51. 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.

52. 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

53. Example 3: Proving Triangles Are Similar
Given: E(–2, –6), F(–3, –2), G(2, –2), H(–4, 2), and J(6, 2).
Prove: ∆EHJ ~ ∆EFG.
Step 1 Plot the points and draw the triangles.

54. Example 3 Continued
Step 2 Use the Distance Formula to find the side lengths.

55. Example 3 Continued
Step 3 Find the similarity ratio.
= 2
= 2
Since and E  E, by the Reflexive Property, ∆EHJ ~ ∆EFG by SAS ~ .

56. Example 4: Using the SSS Similarity Theorem
Graph the image of ∆ABC after a dilation with scale factor
Verify that ∆A'B'C' ~ ∆ABC.

57. Example 4 Continued
Step 1 Multiply each coordinate by to find the coordinates of the vertices of ∆A’B’C’.

58. Example 4 Continued Step 2 Graph ∆A’B’C’. B’ (2, 4) A’ (0, 2)

59. Example 4 Continued
Step 3 Use the Distance Formula to find the side lengths.

60. Example 4 Continued
Step 4 Find the similarity ratio.
Since , ∆ABC ~ ∆A’B’C’ by SSS ~.

61. Check It Out! Example 4
Graph the image of ∆MNP after a dilation with scale factor 3.
Verify that ∆M'N'P' ~ ∆MNP.

62. Check It Out! Example 4 Continued
Step 1 Multiply each coordinate by 3 to find the coordinates of the vertices of ∆M’N’P’.

63. Check It Out! Example 4 Continued
Step 2 Graph ∆M’N’P’.

64. Check It Out! Example 4 Continued
Step 3 Use the Distance Formula to find the side lengths.

65. Check It Out! Example 4 Continued
Step 4 Find the similarity ratio.
Since , ∆MNP ~ ∆M’N’P’ by SSS ~.

67. 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)

68. 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 ~.