Altitude to the Hypotenuse Theorem

1. Altitude to the Hypotenuse Theorem - Tomorrow we will use this theorem to prove the Pythagorean Theorem!

2. Altitude to Hypotenuse Theorem: --the alt to hypotenuse forms two smaller right triangles that will be similar to the original

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4. Altitude to Hypotenuse Theorem:--let’s color the smallest triangle, blue

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6. Altitude to Hypotenuse Theorem:-- next color the middle triangle red

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8. Altitude to Hypotenuse Theorem:--now let’s move & rotate the two small triangles to study all three at the same time in the same orientation

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22. Altitude to Hypotenuse Theorem: x c a y h b

23. Altitude to Hypotenuse Theorem: ? ? ? x c a y h b

24. Altitude to Hypotenuse Theorem: a x h x c a y h b

25. Altitude to Hypotenuse Theorem: a x ? h ? x ? c a y h b

26. Altitude to Hypotenuse Theorem: a x b h h x y c a y h b

27. Altitude to Hypotenuse Theorem: a x h x c a y h b

28. Altitude to Hypotenuse Theorem b h x y c a y h b

29. Altitude to Hypotenuse Theorem:either leg of the large triangle is the geom mean of • the entire hypotenuse and • the segment of the hyp adjacent to that leg. x a y h b

30. Altitude to Hypotenuse Theorem a x b h h y

31. Altitude to Hypotenuse Theorem: a x b h h x y c a y h b

32. Altitude to Hypotenuse Theorem:--the alt to the hypotenuse is the geometric mean of the two segments of the hypotenuse. x c a y h b

33. x c y a h b Altitude to Hypotenuse Theorem:1. Alt to hyp forms 3 ~ rt triangles2. either leg of the large triangle is the geom mean of the entire hyp and the segment of the hyp adjacent to that leg. and  3. the alt to the hyp is the geom mean of the two segments of the hypotenuse. 

34. Altitude to Hypotenuse Theorem:Sample Problem 1: (use part 2 of theorem)2. either leg of the large triangle is the geom mean of the entire hyp and the segment of the hyp adjacent to that leg. and  x 10 6

35. Altitude to Hypotenuse Theorem:Sample Problem 1: (use part 2 of theorem)2. either leg of the large triangle is the geom mean of the entire hyp and the segment of the hyp adjacent to that leg. and  x 10 6

36. Altitude to Hypotenuse Theorem:Sample Problem 2: (use part 3 of theorem)2. the alt to the hyp is the geom mean of the two segments of the hypotenuse.  4 c y 6

37. Altitude to Hypotenuse Theorem:Sample Problem 2: (use part 3 of theorem)2. the alt to the hyp is the geom mean of the two segments of the hypotenuse.  4 c y 6

38. Altitude to Hypotenuse Theorem:Sample Problem 2: (use part 3 of theorem)2. the alt to the hyp is the geom mean of the two segments of the hypotenuse.  4 c y 6

39. Altitude to Hypotenuse Theorem:Sample Problem 3: Find c and h. x c 6 h 12

40. Altitude to Hypotenuse Theorem:Sample Problem 3: Find h and c. x c 6 h 12

41. Altitude to Hypotenuse Theorem:Sample Problem 3: Find h and c. x c 6 h 12