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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

36. 30°- 60°- 90° Practice
60°
30°
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. 10 53 5 30°- 60°- 90° Practice Hypotenuse = short leg * 2
Long Leg = short leg * 3

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

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

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

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

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

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

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

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

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

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

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

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

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

51. 27 9 3 183 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2
Short Leg = Long leg   3

52. 21 7 3 143 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2
Short Leg = Long leg   3

53. 33 113 223 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2
Short Leg = Long leg   3

54. THE END