Similar Triangles and Proportions Lesson 13.4 Similar Triangles and Proportions pp. 556-559

Objectives: 1. To prove theorems relating altitudes, medians, angle bisectors, perimeters, and areas of pairs of similar triangles. 2. To compute lengths of related segments in similar triangles.

Theorem 13.7 In similar triangles the lengths of the altitudes extending to corresponding sides are in the same ratio as the lengths of the corresponding sides.

A B C X 4 D E F 2 Y = 2, DF AC If ∆ABC ~ ∆DEF and . 2 FY CX then =

Theorem 13.8 In similar triangles the lengths of the medians extending to corresponding sides are in the same ratio as the lengths of the corresponding sides.

EXAMPLE Given ABC ~ XYZ, find AD. 12 X Y Z W 8 6

Theorem 13.9 In similar triangles the lengths of the corresponding angle bisectors from the vertices to the points where they intersect the opposite sides are in the same ratio as the lengths of the corresponding sides of the triangles.

Theorem 13.10 In similar triangles the perimeters of the triangles are in the same ratio as the lengths of the corresponding sides.

Theorem 13.11 In similar triangles the ratio of the areas of the triangles is equal to the square of the ratio of the lengths of corresponding sides.

If the area of DEF is 6 units2, find the area of ABC. 12 C D E 4 F If the area of DEF is 6 units2, find the area of ABC.

Practice: ABC ~ GEF, AD and GH are altitudes. Find AB. 4 B C D A 3 F E H G x - 1 x + 1

Practice: MOP ~ RAG, OK and AT are medians. Find TG. 10 O M K P 8

Practice: TOP ~ BAG, OD and AR are angle bisectors. Find OD. 12 9 6 P O T D G A B R

Practice: ABC ~ DEF,. perimeter of ABC = 144 and Practice: ABC ~ DEF, perimeter of ABC = 144 and perimeter of DEF = 16, AB = 12. Find DE. A B C D F E

Practice: ARP ~ HOT, H, O, and T are midpoints, and the area of HOT is 24 units2. Find the area of ARP. R P A O H T

Homework pp. 558-559

►A. Exercises Find the indicated lengths in exercises 1-10. Assume that ∆ABC ~ ∆MNO. 1. AC = 8; MO = 4; CD = 6; find OP. M P N R O Q A D B F C E

►A. Exercises Find the indicated lengths in exercises 1-10. Assume that ∆ABC ~ ∆MNO. 3. BC = 24; NO = 12; AE = 10; find MQ. M P N R O Q A D B F C E

►A. Exercises Find the indicated lengths in exercises 1-10. Assume that ∆ABC ~ ∆MNO. 5. CD = 9; OP = 3; AE = 12; find MQ. M P N R O Q A D B F C E

►A. Exercises Find the indicated lengths in exercises 1-10. Assume that ∆ABC ~ ∆MNO. 7. FB = 10; RN = 7; perimeter of ∆MNO = 28; find the perimeter of ∆ABC. M P N R O Q A D B F C E

►A. Exercises Find the indicated lengths in exercises 1-10. Assume that ∆ABC ~ ∆MNO. 9. BC = 9; NO = 6; Area of ∆MNO = 20; find the area of ∆ABC. M P N R O Q A D B F C E

►B. Exercises Use the same triangles in exercises 1-10 to answer exercises 11-15. 11. State a proportion involving the medians in the triangles. M P N R O Q A D B F C E

►B. Exercises Use the same triangles in exercises 1-10 to answer exercises 11-15. 13. State a proportion involving the angle bisectors in the triangles. M P N R O Q A D B F C E

►B. Exercises Use the same triangles in exercises 1-10 to answer exercises 11-15. 15. State a proportion involving the perimeters of the triangles. M P N R O Q A D B F C E

■ Cumulative Review Give the name of each shaded figure. Classify each as convex or concave. 23.

■ Cumulative Review Give the name of each shaded figure. Classify each as convex or concave. 24.

■ Cumulative Review Give the name of each shaded figure. Classify each as convex or concave. 25.

■ Cumulative Review Give the name of each shaded figure. Classify each as convex or concave. 26.

■ Cumulative Review Give the name of each shaded figure. Classify each as convex or concave. 27.

■ Cumulative Review Give the name of each shaded figure. Classify each as convex or concave. 28.