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# Special Right Triangles

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### Special Right Triangles

1. 1. Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°
2. 2. 45°- 45°- 90° The 45-45-90 triangle is based on the square with sides of 1 unit. 1 1 1 1
3. 3. 45°- 45°- 90° If we draw the diagonals we form two 45-45-90 triangles. 45° 45° 45° 45° 1 1 1 1
4. 4. 45°- 45°- 90° Using the Pythagorean Theorem we can find the length of the diagonal. 45° 45° 45° 45° 1 1 1 1
5. 5. 45°- 45°- 90° 1 2 + 1 2 = c 2 1 + 1 = c 2 2 = c 2  2 = c 45° 45° 45° 45°  2 1 1 1 1
6. 6. 45°- 45°- 90° Conclusion: the ratio of the sides in a 45-45-90 triangle is 1-1-  2 1 1  2 45° 45°
7. 7. 45°- 45°- 90° Practice 4 4  2 SAME leg*  2 4 45° 45°
8. 8. 45°- 45°- 90° Practice 9 9  2 SAME leg*  2 9 45° 45°
9. 9. 45°- 45°- 90° Practice 2 2  2 SAME leg*  2 2 45° 45°
10. 10. 45°- 45°- 90° Practice  14 SAME leg*  2  7  7 45° 45°
11. 11. 45°- 45°- 90° Practice Now Let's Go Backward
12. 12. 45°- 45°- 90° Practice 3  2 hypotenuse   2 45° 45°
13. 13. 45°- 45°- 90° Practice = 3 3  2  2
14. 14. 45°- 45°- 90° Practice 3  2 hypotenuse   2 3 SAME 3 45° 45°
15. 15. 45°- 45°- 90° Practice 6  2 hypotenuse   2 45° 45°
16. 16. 45°- 45°- 90° Practice = 6 6  2  2
17. 17. 45°- 45°- 90° Practice 6  2 hypotenuse   2 6 SAME 6 45° 45°
18. 18. 45°- 45°- 90° Practice 11  2 hypotenuse   2 45° 45°
19. 19. 45°- 45°- 90° Practice = 11 11  2  2
20. 20. 45°- 45°- 90° Practice 11  2 hypotenuse   2 11 SAME 11 45° 45°
21. 21. 45°- 45°- 90° Practice 8 hypotenuse   2 45° 45°
22. 22. 45°- 45°- 90° Practice = 4  2 8  2  2  2 * = 8  2 2
23. 23. 45°- 45°- 90° Practice 8 hypotenuse   2 4  2 SAME 4  2 45° 45°
24. 24. 45°- 45°- 90° Practice 4 hypotenuse   2 45° 45°
25. 25. 45°- 45°- 90° Practice = 2  2 4  2  2  2 * = 4  2 2
26. 26. 45°- 45°- 90° Practice 4 hypotenuse   2 2  2 SAME 2  2 45° 45°
27. 27. 45°- 45°- 90° Practice 6 Hypotenuse   2 45° 45°
28. 28. 45°- 45°- 90° Practice = 3  2 6  2  2  2 * = 6  2 2
29. 29. 45°- 45°- 90° Practice 6 hypotenuse   2 3  2 SAME 3  2 45° 45°
30. 30. 30°- 60°- 90° The 30-60-90 triangle is based on an equilateral triangle with sides of 2 units. 2 2 2 60 ° 60 °
31. 31. 30°- 60°- 90° The altitude (also the angle bisector and median) cuts the triangle into two congruent triangles. 1 1 30 ° 30 ° 2 2 2 60 ° 60 °
32. 32. 30°- 60°- 90° This creates the 30-60-90 triangle with a hypotenuse a short leg and a long leg. hypotenuse Short Leg Long Leg 30 ° 60 °
33. 33. 30°- 60°- 90° Practice 1 2 We saw that the hypotenuse is twice the short leg. We can use the Pythagorean Theorem to find the long leg. 60° 30°
34. 34. 30°- 60°- 90° Practice 1 2  3 A 2 + B 2 = C 2 A 2 + 1 2 = 2 2 A 2 + 1 = 4 A 2 = 3 A =  3 60° 30°
35. 35. 30°- 60°- 90° Conclusion: the ratio of the sides in a 30-60-90 triangle is 1- 2 -  3  3 1 2 60° 30°
36. 36. 30°- 60°- 90° Practice 4 8 Hypotenuse = short leg * 2 4  3 The key is to find the length of the short side. Long Leg = short leg *  3 60° 30°
37. 37. 30°- 60°- 90° Practice 5 10 Hypotenuse = short leg * 2 5  3 Long Leg = short leg *  3 60° 30°
38. 38. 30°- 60°- 90° Practice 7 14 Hypotenuse = short leg * 2 7  3 Long Leg = short leg *  3 60° 30°
39. 39. 30°- 60°- 90° Practice  3 2  3 Hypotenuse = short leg * 2 3 Long Leg = short leg *  3 60° 30°
40. 40. 30°- 60°- 90° Practice  10 2  10 Hypotenuse = short leg * 2  30 Long Leg = short leg *  3 60° 30°
41. 41. 30°- 60°- 90° Practice Now Let's Go Backward
42. 42. 30°- 60°- 90° Practice 11 22 Short Leg = Hypotenuse  2 11  3 Long Leg = short leg *  3 60° 30°
43. 43. 30°- 60°- 90° Practice 2 4 Short Leg = Hypotenuse  2 2  3 Long Leg = short leg *  3 60° 30°
44. 44. 30°- 60°- 90° Practice 9 18 Short Leg = Hypotenuse  2 9  3 Long Leg = short leg *  3 60° 30°
45. 45. 30°- 60°- 90° Practice 15 30 Short Leg = Hypotenuse  2 15  3 Long Leg = short leg *  3 60° 30°
46. 46. 30°- 60°- 90° Practice 23 46 Hypotenuse = Short Leg * 2 23  3 Short Leg = Long leg   3 60° 30°
47. 47. 30°- 60°- 90° Practice 14 28 Hypotenuse = Short Leg * 2 14  3 Short Leg = Long leg   3 60° 30°
48. 48. 30°- 60°- 90° Practice 16 32 Hypotenuse = Short Leg * 2 16  3 Short Leg = Long leg   3 60° 30°
49. 49. 30°- 60°- 90° Practice 3  3 6  3 Hypotenuse = Short Leg * 2 9 Short Leg = Long leg   3 60° 30°
50. 50. 30°- 60°- 90° Practice 4  3 8  3 Hypotenuse = Short Leg * 2 12 Short Leg = Long leg   3 60° 30°
51. 51. 30°- 60°- 90° Practice 9  3 18  3 Hypotenuse = Short Leg * 2 27 Short Leg = Long leg   3 60° 30°
52. 52. 30°- 60°- 90° Practice 7  3 14  3 Hypotenuse = Short Leg * 2 21 Short Leg = Long leg   3 60° 30°
53. 53. 30°- 60°- 90° Practice 11  3 22  3 Hypotenuse = Short Leg * 2 33 Short Leg = Long leg   3 60° 30°
54. 54. THE END