Similar Right Triangles

1. Similar Right Triangles Chapter 8

2. 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. Compare the triangles. What special property do they share? Explain. Activity: Investigating similar right triangles. Do in pairs or threes

3. 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. Theorem 8.1 (page 518) ∆CBD ~ ∆ABC, ∆ACD ~ ∆ABC, ∆CBD ~ ∆ACD

4. In chapter 7, you learned that two triangles are similar if two of their corresponding angles are congruent. For example ∆PQR ~ ∆STU. Recall that the corresponding side lengths of similar triangles are in proportion. Proportions in right triangles

5. In right ∆ABC, altitude CD is drawn to the hypotenuse, forming two smaller right triangles that are similar to ∆ABC From Theorem 8.1, you know that ∆CBD~∆ACD~∆ABC. C C A A D D B B Using a geometric mean to solve problems They ALL look the same! Similar = same shape, different size

6. Geometric Mean Theorems • Theorem 8.2: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments • Theorem 8.3: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. 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.

7. Geometric Mean Theorems If you want to find the Altitude… use Geometric Mean BD CD = GM CD AD CD is the Geometric Mean of AD and BD

8. Example: Use Geometric Mean to find the Altitude of the Triangle 6 x = x 3 18 = x2 √18 = x √9 ∙ √2 = x 3 √2 = x

9. A B Geometric Mean Theorems If you want to find the side on the right side of the triangle… use Geometric Mean GM AB CB = CB DB CB is the Geometric Mean of DB and AB

10. Y is the leg on the right side Rotate the triangle to help visualize what part you need √14 = y Example: Find y (the right leg of the triangle) using Geometric Mean 5 + 2 y = y 2 7 y = y 2 14 = y2

11. A B Geometric Mean Theorems If you want to find the side on the LEFT side of the triangle… use Geometric Mean GM AB AC = AC AD AC is the Geometric Mean of AD and AB

12. Y is the leg on the Left side √245 = y Example: Find y (the left leg of the triangle) using Geometric Mean 35 35 y = y 7 7 245 = y2 y 7√5 = y

13. 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. Ex. 1: Finding the Height of a Roof

14. You may find it helpful to sketch the three similar triangles so that the corresponding angles and sides have the same orientation. Mark the congruent angles. Notice that some sides appear in more than one triangle. For instance XY is the hypotenuse in ∆XYW and the shorter leg in ∆XZY. Solution: ∆XYW ~ ∆YZW ~ ∆XZY.

15. Solution for b. • Use the fact that ∆XYW ~ ∆XZY to write a proportion. YW XY Corresponding side lengths are in proportion. = ZY XZ h 3.1 = Substitute values. 5.5 6.3 6.3h = 5.5(3.1) Cross Product property Solve for unknown h. h≈ 2.7 The height of the roof is about 2.7 meters.