math 9

1. Recall:
a b c
1) 8 15
2) 12 20
3) 24
25

2. Recall:
a b c
1) 8 15 17
2) 12 16 20
3) 7 24
25

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

4. 45°- 45°- 90°
The 45-45-90
triangle is
based on the
square with
sides of 1 unit.
1
1
1
1

5. 45°- 45°- 90°
If we draw the
diagonals we
form two
45-45-90
triangles.
1
1
1
1
45°
45°
45°
45°

6. 45°- 45°- 90°
Using the
Pythagorean
Theorem we can
find the length of
the diagonal.
1
1
1
1
45°
45°
45°
45°

7. 45°- 45°- 90°
12 + 12 = c2
1 + 1 = c2
2 = c2
2 = c
1
45°
1
45°
1
45°
2 1
45°

8. 45°- 45°- 90°
Conclusion:
the ratio of
the sides in a
45-45-90
triangle is
1-1-2
1
1 2
45°
45°

9. 45°- 45°- 90° Practice
4
4 2
SAME leg*2
45°
45°
4

10. 45°- 45°- 90° Practice
9
9 2
SAME leg*2
45°
45°
9

11. 45°- 45°- 90° Practice
2
2 2
SAME leg*2
45°
45°
2

12. 45°- 45°- 90° Practice
14
SAME leg*2
7
45°
45°
7

13. 45°- 45°- 90° Practice

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

15. 45°- 45°- 90° Practice
3 2
2
= 3

16. 45°- 45°- 90° Practice
3 2
hypotenuse
45°
45°
3
SAME
3
2

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

18. 45°- 45°- 90° Practice
6 2
2
= 6

19. 45°- 45°- 90° Practice
6 2
hypotenuse
45°
45°
6
SAME
6
2

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

21. 45°- 45°- 90° Practice
2
11 2
= 11

22. 45°- 45°- 90° Practice
112
hypotenuse
45°
45°
11
SAME
11
2

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

24. 45°- 45°- 90° Practice
8
2
*
2 = 82 = 42
2 2

25. 45°- 45°- 90° Practice
8
45°
45°
42
SAME
42
hypotenuse
2

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

27. 45°- 45°- 90° Practice
4
2
*
2 = 42 = 22
2 2

28. 45°- 45°- 90° Practice
4
hypotenuse
45°
45°
22
SAME
22
2

29. 45°- 45°- 90° Practice
6
Hypotenuse
45°
45°
2

30. 45°- 45°- 90° Practice
6
2
*
2 = 62 = 32
2 2

31. 45°- 45°- 90° Practice
6
hypotenuse
2
45°
45°
32
SAME
32

32. 30°- 60°- 90°
The 30-60-90
triangle is based
on an equilateral
triangle with
sides of 2 units.
2
2
2
60° 60°

33. 2
2
2
60° 60°
30°- 60°- 90°
The altitude (also
the angle bisector
and median) cuts
the triangle into
two congruent
triangles.
1
1
30 °
°30

34. 30°
60°
triangle with a
hypotenuse a
30°- 60°- 90°
This creates
the 30-60-90
Short Leg short leg and
a long leg.
Long
Leg

35. 60°
30°
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.

36. 60°
30°
30°- 60°- 90° Practice
1
2
3
A2 + B2 = C2
A2 + 12 = 22
A2 + 1 = 4
A2 = 3
A = 3

37. 30°- 60°- 90°
Conclusion:
the ratio of
the sides in a
30-60-90
triangle is
1- 2 - 3
60°
30°
3
1
2

38. 60°
30°
30°- 60°- 90° Practice
8
Hypotenuse =
short leg * 2
43
The key is to find
the length of the
short side.
4
Long Leg =
short leg * 3

39. 30°
30°- 60°- 90° Practice
60°
5
Long Leg =
short leg * 3
10 Hypotenuse =
short leg * 2
53

40. 30°
30°- 60°- 90° Practice
60°
7
Long Leg =
short leg * 3
14 Hypotenuse =
short leg * 2
73

41. 30°
30°- 60°- 90° Practice
60°
3
Long Leg =
short leg * 3
23Hypotenuse =
short leg * 2
3

42. 30°
30°- 60°- 90° Practice
210
Hypotenuse =
short leg * 2
30
60°
10
Long Leg =
short leg * 3

43. 30°- 60°- 90° Practice

44. 30°
30°- 60°- 90° Practice
22 Short Leg =
Hypotenuse  2
113
60°
11
Long Leg =
short leg * 3

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

46. 30°
30°- 60°- 90° Practice
18 Short Leg =
Hypotenuse  2
93
60°
9
Long Leg =
short leg * 3

47. 30°
30°- 60°- 90° Practice
30 Short Leg =
Hypotenuse  2
153
60°
15
Long Leg =
short leg * 3

48. 30°
30°- 60°- 90° Practice
46 Hypotenuse =
Short Leg * 2
233
60°
23
Short Leg =
Long leg   3

49. 30°
30°- 60°- 90° Practice
28 Hypotenuse =
Short Leg * 2
143
60°
14
Short Leg =
Long leg   3

50. 60°
30°
30°- 60°- 90° Practice
16
32 Hypotenuse =
Short Leg * 2
163
Short Leg =
Long leg   3

51. 30°
30°- 60°- 90° Practice
63
Hypotenuse =
Short Leg * 2
9
60°
3 3
Short Leg =
Long leg   3

52. 30°
30°- 60°- 90° Practice
83
Hypotenuse =
Short Leg * 2
12
60°
4 3
Short Leg =
Long leg   3

53. 30°
30°- 60°- 90° Practice
183
Hypotenuse =
Short Leg * 2
27
60°
9 3
Short Leg =
Long leg   3

54. 30°
30°- 60°- 90° Practice
143
Hypotenuse =
Short Leg * 2
21
60°
7 3
Short Leg =
Long leg   3

55. 60°
30°
A
B C