1. GEOMETRY 10-3 Similarity in Right Triangles Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

2. GEOMETRY 10-3 Similarity in Right Triangles Warm Up 1. Write a similarity statement comparing the two triangles. Simplify. 2.3. Solve each equation. 4.5. 2x 2 = 50 ∆ADB ~ ∆EDC ±5

3. GEOMETRY 10-3 Similarity in Right Triangles Use geometric mean to find segment lengths in right triangles. Apply similarity relationships in right triangles to solve problems. Objectives

4. GEOMETRY 10-3 Similarity in Right Triangles geometric mean Vocabulary

5. GEOMETRY 10-3 Similarity in Right Triangles In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right triangles.

6. GEOMETRY 10-3 Similarity in Right Triangles

7. GEOMETRY 10-3 Similarity in Right Triangles Example 1: Identifying Similar Right Triangles Write a similarity statement comparing the three triangles. Sketch the three right triangles with the angles of the triangles in corresponding positions. ∆UVW ~ ∆UWZ ~ ∆WVZ. Z W

8. GEOMETRY 10-3 Similarity in Right Triangles TEACH! Example 1 Write a similarity statement comparing the three triangles. Sketch the three right triangles with the angles of the triangles in corresponding positions. ∆LJK ~ ∆JMK ~ ∆LMJ.

9. GEOMETRY 10-3 Similarity in Right Triangles Consider the proportion. In this case, the means of the proportion are the same number, and that number is the geometric mean of the extremes. The geometric mean of two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such that, or x 2 = ab.

10. GEOMETRY 10-3 Similarity in Right Triangles Example 2A: Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 4 and 25 Let x be the geometric mean. x 2 = (4)(25) = 100Def. of geometric mean x = 10Find the positive square root.

11. GEOMETRY 10-3 Similarity in Right Triangles Example 2B: Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. Let x be the geometric mean. 5 and 30 x 2 = (5)(30) = 150 Def. of geometric mean Find the positive square root.

12. GEOMETRY 10-3 Similarity in Right Triangles TEACH! Example 2a Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 2 and 8 Let x be the geometric mean. x 2 = (2)(8) = 16Def. of geometric mean x = 4Find the positive square root.

13. GEOMETRY 10-3 Similarity in Right Triangles TEACH! Example 2b Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. Let x be the geometric mean. 10 and 30 x 2 = (10)(30) = 300Def. of geometric mean Find the positive square root.

14. GEOMETRY 10-3 Similarity in Right Triangles TEACH! Example 2c Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. Let x be the geometric mean. 8 and 9 x 2 = (8)(9) = 72Def. of geometric mean Find the positive square root.

15. GEOMETRY 10-3 Similarity in Right Triangles You can use Theorem 8-1-1 to write proportions comparing the side lengths of the triangles formed by the altitude to the hypotenuse of a right triangle. All the relationships in red involve geometric means.

16. GEOMETRY 10-3 Similarity in Right Triangles

17. GEOMETRY 10-3 Similarity in Right Triangles Example 3: Finding Side Lengths in Right Triangles Find x, y, and z. 6 2 = (9)(x) 6 is the geometric mean of 9 and x. x = 4 Divide both sides by 9. y 2 = (4)(13) = 52 y is the geometric mean of 4 and 13. Find the positive square root. z 2 = (9)(13) = 117 z is the geometric mean of 9 and 13. Find the positive square root.

18. GEOMETRY 10-3 Similarity in Right Triangles Once you’ve found the unknown side lengths, you can use the Pythagorean Theorem to check your answers. Helpful Hint

19. GEOMETRY 10-3 Similarity in Right Triangles TEACH! Example 3 Find u, v, and w. w 2 = (27 + 3)(27) w is the geometric mean of u + 3 and 27. 9 2 = (3)(u) 9 is the geometric mean of u and 3. u = 27 Divide both sides by 3. Find the positive square root. v 2 = (27 + 3)(3) v is the geometric mean of u + 3 and 3. Find the positive square root.

20. GEOMETRY 10-3 Similarity in Right Triangles Example 4: Measurement Application To estimate the height of a Douglas fir, Jan positions herself so that her lines of sight to the top and bottom of the tree form a 90º angle. Her eyes are about 1.6 m above the ground, and she is standing 7.8 m from the tree. What is the height of the tree to the nearest meter?

21. GEOMETRY 10-3 Similarity in Right Triangles Example 4 Continued Let x be the height of the tree above eye level. x = 38.025 ≈ 38 (7.8) 2 = 1.6x The tree is about 38 + 1.6 = 39.6, or 40 m tall. 7.8 is the geometric mean of 1.6 and x. Solve for x and round.

22. GEOMETRY 10-3 Similarity in Right Triangles TEACH! Example 4 A surveyor positions himself so that his line of sight to the top of a cliff and his line of sight to the bottom form a right angle as shown. What is the height of the cliff to the nearest foot?

23. GEOMETRY 10-3 Similarity in Right Triangles TEACH! Example 4 Continued The cliff is about 142.5 + 5.5, or 148 ft high. Let x be the height of cliff above eye level. (28) 2 = 5.5x 28 is the geometric mean of 5.5 and x. Divide both sides by 5.5. x  142.5

24. GEOMETRY 10-3 Similarity in Right Triangles Lesson Quiz: Part I Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 1. 8 and 18 2. 6 and 15 12

25. GEOMETRY 10-3 Similarity in Right Triangles Lesson Quiz: Part II For Items 3–6, use ∆RST. 3. Write a similarity statement comparing the three triangles. 4. If PS = 6 and PT = 9, find PR. 5. If TP = 24 and PR = 6, find RS. 6. Complete the equation (ST) 2 = (TP + PR)(?). ∆RST ~ ∆RPS ~ ∆SPT 4 TP