Geometry Similarity in Right Triangles

1. Geometry Similarity in Right Triangles CONFIDENTIAL

2. Warm up Find the x- intercept and y-intercept for each equation. 1). 3y + 4 = 6x 2). x + 4 = 2y 3). 3y – 15 = 15x 1) x- intercept= 2/3 y-intercept =-4/3 2) x- intercept= -4 y-intercept =2 3) x- intercept= 5 y-intercept =-1 CONFIDENTIAL

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. Theorem 1.1 The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. ∆ABC ~ ∆ACD ~ ∆CBD CONFIDENTIAL

4. Theorem1.2 Given: ∆ABC is a right triangle with altitude CD. Prove: ∆ABC ~ ∆ACD ~ ∆CBD Proof: The right angles in ∆ABC, ∆ACD, and ∆CBD Are all congruent. By the Reflexive Property of Congruence, A ≅ A. Therefore ∆ABC ~ ∆ACD by the AA Similarity Theorem. Similarly, B ≅ B, so ∆ABC ~ ∆CBD. By the Transitive Property of Similarity, ∆ABC~∆ACD~∆CBD. CONFIDENTIAL

5. 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. CONFIDENTIAL

6. Consider the proportion a = x. 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 x = √ab, or x2=ab. x b CONFIDENTIAL

7. Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. A ) 4 and 9 Let x be the geometric mean. Def. of geometric mean Find the positive square root B ) 6 and 15 Let x be the geometric mean. Def. of geometric mean Find the positive square root CONFIDENTIAL

8. Now you try! Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 1a) 2 and 8 1b) 10 and 30 1c) 8 and 9 1a) 4 1b) 10√3 1c) 6√2 CONFIDENTIAL

9. Theorem 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. CONFIDENTIAL

10. Corollaries COROLLARY EXAMPLE DIAGRAM 1.2 The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of two segments of the hypotenuse. 1.3 The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. CONFIDENTIAL

11. Finding Side Lengths in right Triangles Find x, y, and z. X is the geometric mean of 2 and 10. Find the positive square root. Y is the geometric mean of 12 and 10. Find the positive square root. Z is the geometric mean of 12 and 2. Find the positive square root. CONFIDENTIAL

12. Now you try! 2) Find u, v, and w. 2) u = 27, v = 3√10, w = 9√10 CONFIDENTIAL

13. Measurement Application To estimate the height of Big Tex at the State Fair of Texas, Michael steps away from the statue until his line of sight to the top of the status and his line of sight to the bottom of the statue form a 90˚ angle. His eyes are 5 ft above the ground, and he is standing 15 ft 3 in. from Big Tex. How tall is Big Tex to the nearest foot? Let x be the height of Big Tex above eye level. 15 ft 3 in. = 15.25 ft (15.25) = 5x X = 46.5125 = 47 Big Tex is about 47 + 5, or 52 ft tall. Convert 3 in. to 0.25 ft. 15.25 is the geometric mean of 5 and x. Solve for x and round. CONFIDENTIAL

14. Now you try! 3) A surveyor positions himself so that his line of sight to the top of a cliff and his line of sight to the bottom from a right angle as shown. What is the height of the cliff to the nearest foot? 3) 148 ft 5.5 ft CONFIDENTIAL

15. Now some problems for you to practice ! CONFIDENTIAL

16. Assessment Write a similarity statement comparing the three triangles in each diagram. 2) 1) CONFIDENTIAL

17. Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 3). 2 and 50 4). 9 and 12 5). ½ and 8 3) 10 4) 6√3 5) 2 CONFIDENTIAL

18. Find x, y, and z. 6). 7). 6) x = 2√15, y = 2√6, z = 2√10 7) x = 5, y = 10√5, z = 5√5 CONFIDENTIAL

19. 8) Measurement To estimate the length of the US Constitution in Boston harbor, a student located points T and U as shown. What is RS to the nearest tenth? 8) 16√15 CONFIDENTIAL

20. Let’s review 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. Theorem 1.1 The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. ∆ABC ~ ∆ACD ~ ∆CBD CONFIDENTIAL

21. Theorem1.2 Given: ∆ABC is a right triangle with altitude CD. Prove: ∆ABC ~ ∆ACD ~ ∆CBD Proof: The right angles in ∆ABC, ∆ACD, and ∆CBD Are all congruent. By the Reflexive Property of Congruence, A ≅ A. Therefore ∆ABC ~ ∆ACD by the AA Similarity Theorem. Similarly, B ≅ B, so ∆ABC ~ ∆CBD. By the Transitive Property of Similarity, ∆ABC~∆ACD~∆CBD. CONFIDENTIAL

22. 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. CONFIDENTIAL

23. Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. A ) 4 and 9 Let x be the geometric mean. Def. of geometric mean Find the positive square root B ) 6 and 15 Let x be the geometric mean. Def. of geometric mean Find the positive square root CONFIDENTIAL

24. Theorem 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. CONFIDENTIAL

25. Corollaries COROLLARY EXAMPLE DIAGRAM 1.2 The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of two segments of the hypotenuse. 1.3 The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. CONFIDENTIAL

26. Finding Side Lengths in right Triangles Find x, y, and z. X is the geometric mean of 2 and 10. Find the positive square root. Y is the geometric mean of 12 and 10. Find the positive square root. Z is the geometric mean of 12 and 2. Find the positive square root. CONFIDENTIAL

27. Measurement Application To estimate the height of Big Tex at the State Fair of Texas, Michael steps away from the statue until his line of sight to the top of the status and his line of sight to the bottom of the statue form a 90˚ angle. His eyes are 5 ft above the ground, and he is standing 15 ft 3 in. from Big Tex. How tall is Big Tex to the nearest foot? Let x be the height of Big Tex above eye level. 15 ft 3 in. = 15.25 ft (15.25) = 5x X = 46.5125 = 47 Big Tex is about 47 + 5, or 52 ft tall. Convert 3 in. to 0.25 ft. 15.25 is the geometric mean of 5 and x. Solve for x and round. CONFIDENTIAL

28. You did a great job today! CONFIDENTIAL