1. Objective: Use proportions to identify similar polygons

2. EXAMPLE 1
Use similarity statements
In the diagram, ∆RST ~ ∆XYZ
a. List all pairs of congruent
angles.
b. Check that the ratios of
corresponding side lengths are equal. c.
Write the ratios of the corresponding side
lengths in a statement of proportionality.

3. EXAMPLE 1
Use similarity statements
SOLUTION a. R = ~ X, S Y T Z
and RS XY = 20 12 5 3 b. ; ST = 30 18 5 3 YZ ; TR ZX = 25 15 5 3
c. Because the ratios in part (b) are equal, YZ RS XY = ST TR ZX .

4. GUIDED PRACTICE
for Example 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 L R
and JK PQ KL QR LJ RP ;
ANSWER

5. EXAMPLE 2
Find the scale factor
Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of ZYXW to FGHJ.

6. EXAMPLE 2
Find the scale factor
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.

7. EXAMPLE 2
Find the scale factor
SOLUTION
STEP 2
Show that corresponding side lengths are proportional. ZY FG 25 20 = 5 4 = YX GH 30 24 = 5 4 = XW HJ 15 12 = = 5 4 WZ JF 20 16 = = 5 4

8. EXAMPLE 2
Find the scale factor
SOLUTION
The ratios are equal, so the corresponding side lengths are proportional.
So ZYXW ~ FGHJ. The scale factor of ZYXW to
FGHJ is
ANSWER 5 4 .

9. EXAMPLE 3
Use similar polygons
In the diagram, ∆DEF ~ ∆MNP. Find the value of x.
ALGEBRA

10. EXAMPLE 3 Use similar polygons SOLUTION
The triangles are similar, so the corresponding side lengths are proportional. MN DE NP EF =
Write proportion. = 12 9 20 x
Substitute.
12x = 180
Cross Products Property
x = 15
Solve for x.

11. GUIDED PRACTICE
for Examples 2 and 3
In the diagram, ABCD ~ QRST.
2. What is the scale factor of QRST to ABCD ? 1 2
ANSWER
Find the value of x.
ANSWER 8

12. EXAMPLE 4
Find perimeters of similar figures
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.

13. EXAMPLE 4
Find perimeters of similar figures
Find the perimeter of an Olympic pool and the new pool. b.
SOLUTION
Because the new pool will be similar to an Olympic pool, the scale factor is the ratio of the lengths, a. 40 50 = 4 5

14. Find perimeters of similar figures
EXAMPLE 4
Find perimeters of similar figures
The perimeter of an Olympic pool is (50) + 2(25) = 150 meters. You can use Theorem 6.1 to find the perimeter x of the new pool. b. x
150 4 5 =
Use Theorem 6.1 to write a proportion.
x = 120
Multiply each side by 150 and simplify.
The perimeter of the new pool is 120 meters.
ANSWER

15. GUIDED PRACTICE
for Example 4
In the diagram, ABCDE ~ FGHJK.
Find the scale factor of FGHJK to ABCDE. 3 2
ANSWER
5. Find the value of x.
ANSWER 12
6. Find The perimeter of ABCDE.
ANSWER 46

16. EXAMPLE 5
Use a scale factor
In the diagram, ∆TPR ~ ∆XPZ. Find the length of the altitude PS .
SOLUTION
First, find the scale factor of ∆TPR to ∆XPZ. TR XZ
6 + 6 =
8 + 8 = 12 16 = 3 4

17. The length of the altitude PS is 15. ANSWER
EXAMPLE 5
Use a scale factor
Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion. PS PY 3 4 =
Write proportion. PS 20 3 4 =
Substitute 20 for PY. = PS 15
Multiply each side by 20 and simplify.
The length of the altitude PS is 15.
ANSWER

18. GUIDED PRACTICE
for Example 5
In the diagram, ∆JKL ~ ∆ EFG. Find the length of the median KM. 7.
ANSWER 42