7.5 Proportions and Similar Triangles

[Pages:7]Page 1 of 7

7.5 Proportions and

Similar Triangles

Goal

Use the Triangle Proportionality Theorem and its converse.

Key Words

? midsegment of a triangle

Geo-Activity Investigating Proportional Segments

1 Draw a triangle. Label its vertices

A, B, and C. Make sure that each

side is at least 4 cm. Draw a point on A&B*. Label the point D.

2 Draw a line through D parallel

to A&C*. Label the intersection of

the line and B&C* as point E.

B D

A

B D

E

A

C

C

3 Measure B&D*, D&A*, B&E*, and E&C* in centimeters. Then calculate the

ratios DBDA and EBCE .

4 Make a conjecture about the ratios of segment lengths of a

triangle's sides when the triangle is cut by a line parallel to the triangle's third side.

Proportionality Suppose that a point P lies on G&H** and a point Q lies on

J&K*.

If

GP PH

JQ QK

,

then

we

say

that

G&H*

and

&JK*

are

divided

proportionally.

G 3P

6

H

J 4P

8

K

In the Geo-Activity above, D&E* divides A&B* and C&B* proportionally.

THEOREM 7.4

Triangle Proportionality Theorem

Words If a line parallel to one side of

a triangle intersects the other two sides, then it divides the two sides proportionally.

T P

Symbols

In

T

QRS,

if

T&U*

Q&S*,

then

RT TQ

RU US

.

R

U S

386 Chapter 7 Similarity

IStudent Help



MORE EXAMPLES More examples at

Page 2 of 7

EXAMPLE 1 Find Segment Lengths

Find the value of x.

C

4x

D

E

8

12

B

A

Solution CD CE

DB EA 48 1x2

4 p 12 8 p x 48 8x 488 88x 6x

Triangle Proportionality Theorem

Substitute 4 for CD, 8 for DB, x for CE, and 12 for EA. Cross product property Multiply. Divide each side by 8. Simplify.

EXAMPLE 2 Find Segment Lengths

Find the value of y.

R 9 P3

S

T yP

20

Solution

You know that PS 20 and PT y. By the Segment Addition Postulate, TS 20 y.

PQ QR

PT TS

3

y

9 20 y

3(20 y) 9 p y

60 3y 9y

60 3y 3y 9y 3y

60 12y

6102 1122y

5y

Triangle Proportionality Theorem

Substitute 3 for PQ, 9 for QR, y for PT, and (20 y) for TS. Cross product property Distributive property Add 3y to each side. Simplify.

Divide each side by 12.

Simplify.

7.5 Proportions and Similar Triangles 387

Page 3 of 7

THEOREM 7.5

Converse of the Triangle Proportionality Theorem

Words If a line divides two sides of a triangle

proportionally, then it is parallel to the third side.

T

Symbols

In

TQRS,

if

RT TQ

RU US

,

then

T&U*

Q&S*.

P

R

U S

EXAMPLE 3 Determine Parallels

Given the diagram, determine whether MxxxxNxx is parallel to GxxHxxx.

G 21 M

56

N 16 H

L

48

Solution Find and simplify the ratios of the two sides divided by MxxxxNxx.

LM 56 8 MG 21 3

LN 48 3 NH 16 1

ANSWER

Because

8 3

31, MxxxxNxx is not parallel to GxxHxxx.

Find Segment Lengths and Determine Parallels

Find the value of the variable.

1.

4

x

2. y

14

5

10

15 6

Given the diagram, determine whether Q&R* is parallel to S&T*. Explain.

3. T

21

R 15

P

P 17 23 S

4. P

4P 6

R

12 8

S

T

388 Chapter 7 Similarity

Page 4 of 7

A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle. The following theorem about midsegments is a special case of the Triangle Proportionality Theorem.

THEOREM 7.6

The Midsegment Theorem

Words The segment connecting the midpoints

of two sides of a triangle is parallel to the third side and is half as long.

C D

Symbols In T ABC, if CD DA and CE EB,

then D&E* A&B* and DE 12AB.

A

E B

EXAMPLE 4 Use the Midsegment Theorem

Find the length of Q&S*.

R

P

S

Solution

P

10

T

From the marks on the diagram, you know S is the midpoint of R&T*, and Q is the midpoint of R&P*. Therefore, Q&S* is a midsegment of T PRT. Use the Midsegment Theorem to write the following

equation.

QS 12PT 12(10) 5

ANSWER The length of Q&S* is 5.

Use the Midsegment Theorem Find the value of the variable.

5. p

16

6.

q

8

14

11

8

11

7. Use the Midsegment Theorem to find

A

the perimeter of T ABC.

3

4

5

3

B

4 C

7.5 Proportions and Similar Triangles 389

Page 5 of 7

7.5 Exercises

Guided Practice

Vocabulary Check

Complete the statement.

1. The __?__ Theorem states that if a line divides two sides of a triangle proportionally, then it is __?__ to the third side.

2. A __?__ of a triangle is a segment that connects the midpoints of two sides of a triangle.

Skill Check Copy and complete the proportion using the diagram below.

3. DADB E?C

5.

AD ?

AE AC

4. D?A ECAE

6.

BD BA

CE ?

A D B

E C

Find the value of the variable.

7.

8.

4

9.

z

8

x

10 y

4

15

2

3

Practice and Applications

Extra Practice

See p. 688.

Using Algebra Solve the proportion.

10. 2 m 3 36

11. t 5 2 12

12. 21 7 y 18

13. 27 3 r4

Finding Segment Lengths Find the value of the variable.

14. 7

9

x

18

15.

21

y

7

6

16. 15

p

6

20

Homework Help

Example 1: Exs. 14?19 Example 2: Exs. 14?19 Example 3: Exs. 20?23 Example 4: Exs. 24?29,

33?37

17. 5 q

15

24

18.

20

24 c

12 19.

z 12 55

32

390 Chapter 7 Similarity

Page 6 of 7

Fractals

Determining Parallels Given the diagram, determine whether Q&S* is parallel to P&T*.

20.

R

21.

5T

S

16

8

12.5

P 4

S 2

P

T

R

10

P4 P

22. T P

34

S 15

35

P 16 R

23.

R

12 P 9

P

20 S 15

T

Using the Midsegment Theorem Find the value of the variable.

24. x

25.

a

11

26. 18

b

4

27.

7

7

c

5 8

8

28. y 1

29. 10 10

19 z

19 27

Visualize It! The design below approximates a fractal. Begin with

an equilateral triangle. Shade the triangle formed by the three midsegments. Continue the process for each unshaded triangle.

16

16

16

FRACTALS are shapes that look the same at many levels of magnification. Take a small part of the image above and you will see that it looks similar to the whole image.

Application Links



Stage 0

Stage 1

Stage 2

Stage 3

30. Find the perimeter of the dark blue triangle in Stage 1.

31. Challenge Find the total perimeter of all the dark blue triangles in Stage 2.

32. Challenge Find the total perimeter of all the dark blue triangles in Stage 3.

7.5 Proportions and Similar Triangles 391

Page 7 of 7

Midsegment Theorem Use the diagram below to complete

the statement.

B

33. LxxMxxx __?__

34. A&B* __?__

35. If AC 15, then LN __?__. 36. If MN 7.4, then AB __?__.

L

N

37. If NC 9.5, then LM __?__.

A

M

C

Technology In Exercises 38 and 39, use geometry software to complete the steps below.

1 Draw T ABC. 2 Construct the angle bisector of aA.

3 Construct the intersection of the

angle bisector and B&C*. Label it D.

B D

A

4 Measure D&C**, A&C*, D&B*, and A&B*. Then

C

calculate the ratios DACC and DABB.

38. Drag one or more of the triangle's vertices. What do you notice about the ratios as the shape changes?

39. Complete the conjecture: If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are __?__ to the lengths of the other two sides.

Standardized Test 40. Multiple Choice What is the value of x?

Practice

A 21

B 24

8

C 32

D 42

x

42

16

Mixed Review

Reflections Determine if the entire word has any lines of symmetry. If so, write the word and draw the line(s) of symmetry. (Lesson 5.7)

41.

42.

43.

Algebra Skills

Finding Slope Find the slope of the line that passes through the points. (Skills Review, p. 665)

44. (0, 2) and (4, 8)

45. (1, 2) and (3, 4)

46. (5, 2) and (5, 3)

47. (5, 6) and (1, 2) 48. (4, 4) and (2, 0) 49. (3, 7) and (1, 3)

50. (5, 3) and (1, 1) 51. (0, 4) and (3, 5) 52. (3, 2) and (6, 5)

392 Chapter 7 Similarity

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download