9.1 Similar Right Triangles

1. 9.1 Similar Right Triangles Geometry Mrs. Blanco

2. Proportions in right triangles • In Chapter 8, you learned that two triangles are similar if two of their corresponding angles are congruent. • For example∆PQR ~ ∆STU.

3. Activity: Investigating similar right triangles. Do in pairs or threes • Cut an index card along one of its diagonals. • On one of the right triangles, draw an altitude from the right angle to the hypotenuse. Cut along the altitude to form two right triangles. • You should now have three right triangles. Meausure all of the angles on each of the triangles. Compare the triangles. What special property do they share? Explain.

4. Theorem 9.1 • If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. ∆CBA ~ ∆DCA~∆DBC

5. Ex. 1: Finding the Height of a Roof • Roof Height. A roof has a cross section that is a right angle. The diagram shows the approximate dimensions of this cross section. • A. Identify the similar triangles. • B. Find the height, h, of the roof. NEXT SLIDE

6. Solution for b. • Use the fact that ∆XYW ~ ∆XZY to write a proportion. YW XY = ZY XZ h 3.1 = 5.5 6.3 6.3h = 5.5(3.1) h≈ 2.7 The height of the roof is about 2.7 meters.

7. Be Careful!!!! BD CD = CD AD AB CB = CB DB AB AC = AC AD

8. In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. Theorem 9.2: The length of the altitude is the geometric mean of the lengths of the two segments Geometric Mean Theorems BD CD = CD AD

9. Geometric Means Theorem • Theorem 9.3: The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. AB CB = CB DB AB AC = AC AD

10. Apply the Geometric means theorems. Find x and y!!!!

11. Little More Difficult!!

12. Practice: Page 531-532 #13-30 Every Other Odd 15 minutes