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Two Special Right Triangles

Two Special Right Triangles. 45°- 45°- 90° 30°- 60°- 90°. 1. 1. 1. 1. 45°- 45°- 90°. The 45-45-90 triangle is based on the square with sides of 1 unit. 1. 1. 1. 1. 45°- 45°- 90°. If we draw the diagonals we form two 45-45-90 triangles. 45°. 45°. 45°. 45°. 1. 1. 1. 1.

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Two Special Right Triangles

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  1. Two Special Right Triangles 45°- 45°- 90° 30°- 60°- 90°

  2. 1 1 1 1 45°- 45°- 90° The 45-45-90 triangle is based on the square with sides of 1 unit.

  3. 1 1 1 1 45°- 45°- 90° If we draw the diagonals we form two 45-45-90 triangles. 45° 45° 45° 45°

  4. 1 1 1 1 45°- 45°- 90° Using the Pythagorean Theorem we can find the length of the diagonal. 45° 45° 45° 45°

  5. 1 1 1 1 45°- 45°- 90° 12 + 12 = c2 1 + 1 = c2 2 = c2 2 = c 45° 45° 2 45° 45°

  6. 45° 2 1 45° 1 45°- 45°- 90° Conclusion: the ratio of the sides in a 45-45-90 triangle is 1-1-2

  7. 45° 4 45° 45°- 45°- 90° Practice 4 2 4 SAME leg*2

  8. 45° 9 45° 45°- 45°- 90° Practice 9 2 9 SAME leg*2

  9. 45° 2 45° 45°- 45°- 90° Practice 2 2 2 SAME leg*2

  10. 45° 7 45° 45°- 45°- 90° Practice 14 7 SAME leg*2

  11. 45°- 45°- 90° Practice Now Let's Go Backward

  12. 45° 45° 45°- 45°- 90° Practice 3 2 hypotenuse2

  13. 3 2 2 45°- 45°- 90° Practice = 3

  14. 45° 45° 45°- 45°- 90° Practice 3 2 3 3 SAME hypotenuse2

  15. 45° 45° 45°- 45°- 90° Practice 6 2 hypotenuse2

  16. 6 2 2 45°- 45°- 90° Practice = 6

  17. 45° 45° 45°- 45°- 90° Practice 6 2 6 6 SAME hypotenuse2

  18. 45° 45° 45°- 45°- 90° Practice 11 2 hypotenuse2

  19. 11 2 2 45°- 45°- 90° Practice = 11

  20. 45° 45° 45°- 45°- 90° Practice 112 11 11 SAME hypotenuse2

  21. 45° 45° 45°- 45°- 90° Practice 8 hypotenuse2

  22. 8 2 82 = * 2 2 2 45°- 45°- 90° Practice = 42

  23. 45° 45° 45°- 45°- 90° Practice 8 42 42 SAME hypotenuse2

  24. 45° 45° 45°- 45°- 90° Practice 4 hypotenuse2

  25. 4 2 42 = * 2 2 2 45°- 45°- 90° Practice = 22

  26. 45° 45° 45°- 45°- 90° Practice 4 22 22 SAME hypotenuse2

  27. 45° 45° 45°- 45°- 90° Practice 6 Hypotenuse 2

  28. 6 2 62 = * 2 2 2 45°- 45°- 90° Practice = 32

  29. 45° 45° 45°- 45°- 90° Practice 6 32 32 SAME hypotenuse2

  30. 2 2 60° 60° 2 30°- 60°- 90° The 30-60-90 triangle is based on an equilateral triangle with sides of 2 units.

  31. 2 2 60° 60° 2 30°- 60°- 90° The altitude (also the angle bisector and median) cuts the triangle into two congruent triangles. 30° 30° 1 1

  32. 30° 60° 30°- 60°- 90° This creates the 30-60-90 triangle with a hypotenuse a short leg and a long leg. Long Leg hypotenuse Short Leg

  33. 30° 60° 30°- 60°- 90° Practice We saw that the hypotenuse is twice the short leg. 2 We can use the Pythagorean Theorem to find the long leg. 1

  34. 30° 60° 30°- 60°- 90° Practice A2 + B2 = C2 A2 + 12 = 22 A2 + 1 = 4 A2 = 3 A = 3 2 3 1

  35. 30° 60° 30°- 60°- 90° Conclusion: the ratio of the sides in a 30-60-90 triangle is 1- 2 - 3 2 3 1

  36. 30° 60° 30°- 60°- 90° Practice The key is to find the length of the short side. 8 43 Hypotenuse = short leg * 2 4 Long Leg = short leg *3

  37. 30° 60° 30°- 60°- 90° Practice 10 Hypotenuse = short leg * 2 53 5 Long Leg = short leg *3

  38. 30° 60° 30°- 60°- 90° Practice 14 Hypotenuse = short leg * 2 73 7 Long Leg = short leg *3

  39. 30° 60° 30°- 60°- 90° Practice 23 Hypotenuse = short leg * 2 3 3 Long Leg = short leg *3

  40. 30° 60° 30°- 60°- 90° Practice 210 30 Hypotenuse = short leg * 2 10 Long Leg = short leg *3

  41. 30°- 60°- 90° Practice Now Let's Go Backward

  42. 30° 60° 30°- 60°- 90° Practice 22 Short Leg = Hypotenuse 2 113 11 Long Leg = short leg *3

  43. 30° 60° 30°- 60°- 90° Practice 4 Short Leg = Hypotenuse 2 23 2 Long Leg = short leg *3

  44. 30° 60° 30°- 60°- 90° Practice 18 Short Leg = Hypotenuse 2 93 9 Long Leg = short leg *3

  45. 30° 60° 30°- 60°- 90° Practice 30 Short Leg = Hypotenuse 2 153 15 Long Leg = short leg *3

  46. 30° 60° 30°- 60°- 90° Practice 46 Hypotenuse = Short Leg * 2 233 23 Short Leg = Long leg 3

  47. 30° 60° 30°- 60°- 90° Practice 28 Hypotenuse = Short Leg * 2 143 14 Short Leg = Long leg 3

  48. 30° 60° 30°- 60°- 90° Practice 32 Hypotenuse = Short Leg * 2 163 16 Short Leg = Long leg 3

  49. 30° 60° 30°- 60°- 90° Practice 63 Hypotenuse = Short Leg * 2 9 3 3 Short Leg = Long leg 3

  50. 30° 60° 30°- 60°- 90° Practice 83 Hypotenuse = Short Leg * 2 12 4 3 Short Leg = Long leg 3

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