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6.3 Similar Polygons

6.3 Similar Polygons. Objective : Use proportions to identify s imilar p olygons. 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

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6.3 Similar Polygons

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  1. 6.3 Similar Polygons Objective: Use proportions to identify similar polygons

  2. 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. EXAMPLE 1 Use similarity statements

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

  4. ANSWER ; ~ ~ ~ J P, L R and Q K = = = JK LJ KL = = RP PQ QR for Example 1 GUIDED PRACTICE 1. Given ∆ JKL ~ ∆ PQR, list all pairs of congruent angles. Write the ratios of the corresponding side lengths in a statement of proportionality.

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

  6. STEP 1 Identify pairs of congruent angles. From the diagram, you can see that Z F, Y G, and X H.Angles Wand J are right angles, so W J.So,the corresponding angles are congruent. EXAMPLE 2 Find the scale factor SOLUTION

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

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

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

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

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

  12. 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. a. Find the scale factor of the new pool to an Olympic pool. EXAMPLE 4 Find perimeters of similar figures

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

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

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

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

  17. = 3 PS 3 4 PY 4 PS = 20 ANSWER = PS 15 The length of the altitude PSis 15. 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. Write proportion. Substitute 20 for PY. Multiply each side by 20 and simplify.

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

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