6.3 use similar polygons

6.36.3 Use Similar Polygons

Bell Thinger

2. The scale of a map is 1 cm : 10 mi. The actual

distance between two towns is 4.3 miles. Find the

length on the map.

ANSWER 0.43 cm

ANSWER 72

1. Solve 1210 = 60

x

3. A model train engine is 9 centimeters long. The

actual engine is 18 meters long. What is the scale of

the model?

ANSWER 1 cm : 2 m

6.3

6.3Example 1

b. Check that the ratios of

corresponding side

lengths are equal.

In the diagram, ∆RST ~ ∆XYZ

a. List all pairs of

congruent angles.

c. Write the ratios of the corresponding side

lengths in a statement of proportionality.

SOLUTION

a. R ≅ X, S ≅ Y and T ≅ Z

6.3

b. Check that the ratios of

corresponding side

lengths are equal.

In the diagram, ∆RST ~ ∆XYZ

a. List all pairs of

congruent angles.

c. Write the ratios of the corresponding side

lengths in a statement of proportionality.

SOLUTION

TRZX =

2515 =

53

RSXY

= 2012

=53

b. ;ST

=3018

=53YZ

;

Example 1

6.3

b. Check that the ratios of

corresponding side

lengths are equal.

In the diagram, ∆RST ~ ∆XYZ

a. List all pairs of

congruent angles.

c. Write the ratios of the corresponding side

lengths in a statement of proportionality.

SOLUTION

c. Because the ratios in part (b) are equal,

YZRSXY

=ST

=TRZX

.

Example 1

6.3Guided Practice

1. Given ∆ JKL ~ ∆ PQR, list all pairs of congruent

angles. Write the ratios of the corresponding side lengths

in a statement of proportionality.

∠J ≅ ∠P, ∠K ≅ ∠Q and ∠L ≅ ∠R ;JKPQ

=KL

QR=

LJRP

ANSWER

6.3Example 2

Determine whether the polygons are similar. If they

are, write a similarity statement and find the scale

factor of ZYXW to FGHJ.

6.3

SOLUTION

STEP 1

Identify pairs of congruent angles. From the

diagram, you can see that ∠Z ≅ ∠F, ∠Y ≅ ∠G, and ∠X ≅ ∠H.

Angles W and J are right angles, so ∠W ≅ ∠J. So, the

corresponding angles are congruent.

Example 2

6.3

SOLUTION

STEP 2

Show that corresponding side lengths are proportional.

ZYFG

2520

=54

=

XWHJ

1512

= =54

WZJF

2016

= =54

YXGH

54

=3024

=

Example 2

6.3

SOLUTION

The ratios are equal, so the corresponding side lengths

are proportional.

So ZYXW ~ FGHJ. The scale factor of ZYXW to

FGHJ is 54

.

Example 2

6.3Example 3

In the diagram,

∆DEF ~ ∆MNP. Find the value

of x.

ALGEBRA

Write proportion.

Substitute.

Cross Products Property

Solve for x.

SOLUTION

The triangles are similar, so the corresponding side

lengths are proportional.

x = 15

12x = 180

MNDE

NPEF

=

=129

20x

6.3Guided Practice

In the diagram, ABCD ~ QRST.

2. What is the scale factor of QRST to ABCD ?

1

2ANSWER

3. Find the value of x.

ANSWER 8

6.3

6.3Example 4

Swimming A town is

building a new swimming

pool. An Olympic pool is

rectangular with length

50 meters and width

25 meters. The new pool

will be similar in

shape, but only 40 meters

long.Find the scale factor of the new pool to an

Olympic pool.

a.

Find the perimeter of an Olympic pool and the

new pool.

b.

6.3

SOLUTION

Because the new pool will be similar to an Olympic

pool, the scale factor is the ratio of the lengths,

a.

40

50=

45

.

x150

45

= Use Theorem 6.1 to write a proportion.

x = 120 Multiply each side by 150 and simplify.

The perimeter of an Olympic pool is

2(50) + 2(25) = 150 meters. You can use Theorem 6.1

to find the perimeter x of the new pool.

b.

The perimeter of the new pool is 120 meters.

Example 4

6.3Guided Practice

4. Find the scale factor of

FGHJK to ABCDE.

In the diagram, ABCDE ~ FGHJK.

3

2ANSWER

5. Find the value of x. ANSWER 12

6. Find The perimeter of ABCDE. ANSWER 48

6.3

6.3Example 5

In the diagram, ∆TPR ~ ∆XPZ. Find the length of the

altitude PS .

SOLUTION

First, find the scale factor of ∆TPR to ∆XPZ.

TRXZ

6 + 6=

8 + 8=

1216

=34

6.3

Because the ratio of the lengths of the altitudes in

similar triangles is equal to the scale factor, you can

write the following proportion.

Write proportion.

Substitute 20 for PY.

Multiply each side by 20 and simplify.

PSPY

34

=

PS20

34

=

=PS 15

The length of the altitude PS is 15.

Example 5

6.3Guided Practice

In the diagram, ∆JKL ~ ∆ EFG. Find the length of

the median KM.

7.

ANSWER 42

6.3Exit Slip

1. Determine whether the polygons are similar. If

they are, write a similarity statement and find the

scale factor of EFGH to KLMN.

ANSWER Yes; EFGH ~ KLMN; the scale factor is

2:1

6.3Exit Slip

2. In the diagram, DEF ~ HJK. Find the value

of x.

ANSWER 13.5

6.3Exit Slip

3. Two similar triangles have the scale factor 5 : 4.

Find the ratio of their corresponding altitudes and

median.

4. Two similar triangles have the scale factor 3 : 7.

Find the ratio of their corresponding perimeters

and areas.

5 : 4 ; 5 : 4ANSWER

ANSWER 3 : 7; 9: 49

6.3

HomeworkPg 392-395

#4, 8, 13, 20, 31