1. Geometry 9.1 Similar Right Triangles

2. June 5, 2016Geometry 9.1 Similar Right Triangles2 Similar Triangles A B C D Remember : If two angles of one triangle are congruent to two angles of a another triangle, then the triangles are similar. (AA Postulate)

3. Similar Right Triangles—If an altitude is drawn from the right angle to the hypotenuse, the two smaller triangles are similar to the original triangle. June 5, 2016Geometry 9.1 Similar Right Triangles3 A B C D  ABC ~  ACD ~  CBD (AA~ Postulate) For clarity, we name the segments. a b mn h c

4. June 5, 2016Geometry 9.1 Similar Right Triangles4 Similar Triangles  ABC ~  ACD ~  CBD A B C D a b mn h c

5. June 5, 2016Geometry 9.1 Similar Right Triangles5 Similar Triangles  ABC ~  ACD ~  CBD B A C D a b mn h c

6. June 5, 2016Geometry 9.1 Similar Right Triangles6 Similar Triangles  ABC ~  ACD ~  CBD B a m h b n h A C D a b mn h c

7. June 5, 2016Geometry 9.1 Similar Right Triangles7 Similar Triangles  ABC ~  ACD ~  CBD B a m h b n h A C a b c

8. June 5, 2016Geometry 9.1 Similar Right Triangles8 Similar Triangles  ABC ~  ACD ~  CBD B a m h b n h A C a b c ? ?

9. June 5, 2016Geometry 9.1 Similar Right Triangles9 Similar Triangles  ABC ~  ACD ~  CBD B a m h b n h A C a b c ?

10. June 5, 2016Geometry 9.1 Similar Right Triangles10 Similar Triangles  ABC ~  ACD ~  CBD B a m h b n h A C a b c ?

11. June 5, 2016Geometry 9.1 Similar Right Triangles11 Similar Triangles  ABC ~  ACD ~  CBD B A C D a b mn h c

12. June 5, 2016Geometry 9.1 Similar Right Triangles12 Similar Triangles  ABC ~  ACD ~  CBD B A C D a b mn h c

13. June 5, 2016Geometry 9.1 Similar Right Triangles13 Similar Triangles  ABC ~  ACD ~  CBD B A C D a b mn h c

14. June 5, 2016Geometry 9.1 Similar Right Triangles14 Similar Triangles  ABC ~  ACD ~  CBD B A C D a b mn h c

15. June 5, 2016Geometry 9.1 Similar Right Triangles15 Similar Triangles  ABC ~  ACD ~  CBD B A C D a b mn h c

16. June 5, 2016Geometry 9.1 Similar Right Triangles16 Geometric Mean These expressions are called geometric means.

17. June 5, 2016Geometry 9.1 Similar Right Triangles17 Means (Averages) Arithmetic mean of x & y: Geometric mean of x & y: 2

18. June 5, 2016Geometry 9.1 Similar Right Triangles18 Theorem 9.2—”The Altitude Rule The altitude drawn to the hypotenuse of a right triangle is the geometric mean of the segments on the hypotenuse. mn h

19. June 5, 2016Geometry 9.1 Similar Right Triangles19 ExampleFind h. 4 9 h 6

20. June 5, 2016Geometry 9.1 Similar Right Triangles20 Your TurnFind x. x 8 4 2

21. June 5, 2016Geometry 9.1 Similar Right Triangles21 Theorem 9.3—”The Leg Rule” The length of each side of a right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that side. a b mn c

22. June 5, 2016Geometry 9.1 Similar Right Triangles22 ExampleFind a & b. a b 510 15

23. June 5, 2016Geometry 9.1 Similar Right Triangles23 Your TurnFind a & b. a b 2 8

24. June 5, 2016Geometry 9.1 Similar Right Triangles24 All you need to know…  ABC ~  ACD ~  CBD D a b mn h c

25. June 5, 2016Geometry 9.1 Similar Right Triangles25 Quickly Complete the Equation x y 6 4 t

26. June 5, 2016Geometry 9.1 Similar Right Triangles26 Quickly Complete the Equation x y 6 4 t

27. June 5, 2016Geometry 9.1 Similar Right Triangles27 Quickly Complete the Equation x y 12 j 5

28. June 5, 2016Geometry 9.1 Similar Right Triangles28 Quickly Complete the Equation x y 7 4 t

29. June 5, 2016Geometry 9.1 Similar Right Triangles29 One more: Solve for a, b, & c. b a 4 6 x c

30. June 5, 2016Geometry 9.1 Similar Right Triangles30 Solution b a 4 6 x c Begin here

31. June 5, 2016Geometry 9.1 Similar Right Triangles31 Solution b a 4 6 9 c 13

32. June 5, 2016Geometry 9.1 Similar Right Triangles32 Solution b 10.82 4 6 9 13

33. June 5, 2016Geometry 9.1 Similar Right Triangles33 Solution 7.21 4 6 9 13 10.82

34. June 5, 2016Geometry 9.1 Similar Right Triangles34 This is hard! No, it isn’t. Ask: what segment do you want to find? Which others do you need to know? Which formula form? Solve the formula for the missing segment.

35. June 5, 2016Geometry 9.1 Similar Right Triangles35 All you need to know…  ABC ~  ACD ~  CBD D a b mn h c

36. June 5, 2016Geometry 9.1 Similar Right Triangles36 You do it. Solve for a, b, h. 21 b a 6 h 15

37. June 5, 2016Geometry 9.1 Similar Right Triangles37 Try this. Solve for a. a 4.5 6 m

38. June 5, 2016Geometry 9.1 Similar Right Triangles38 Try this. Solve for a. 12.5 a 4.5 6 8 Is there another way to solve for “a”? Yeppers! Pythagorean Theorem!!! 6 2 + 8 2 = a 2

39. June 5, 2016Geometry 9.1 Similar Right Triangles39 Is this possible? 5 9 16 No: 9 ≠ 8.94 h