Special Right Triangles

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

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

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

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

5. 45°- 45°- 90°
1
2
+ 1
2
= c
2
1 + 1 = c
2
2 = c
2
Ö2 = c
1
1
1
1
45°
45°
45°
45°
Ö2

6. 45°- 45°- 90°
Conclusion:
the ratio of
the sides in a
45-45-90
triangle is
1-1-Ö2
1
1
Ö2
45°
45°

7. 45°- 45°- 90° Practice
4
4 Ö2
SAME
leg*Ö2
4
45°
45°

8. 45°- 45°- 90° Practice
9
9 Ö2
SAME
leg*Ö2
9
45°
45°

9. 45°- 45°- 90° Practice
2
2 Ö2
SAME
leg*Ö2
2
45°
45°

10. 45°- 45°- 90° Practice
Ö14
SAME
leg*Ö2
Ö7
Ö7
45°
45°

11. 45°- 45°- 90° Practice

12. 45°- 45°- 90° Practice
3 Ö2
hypotenuse
¸Ö2
45°
45°

13. 45°- 45°- 90° Practice
3 Ö2
Ö2
= 3

14. 45°- 45°- 90° Practice
3 Ö2
hypotenuse
¸Ö2
45°
45°
3SAME
3

15. 45°- 45°- 90° Practice
6 Ö2
hypotenuse
¸Ö2
45°
45°

16. 45°- 45°- 90° Practice
6 Ö2
Ö2
= 6

17. 45°- 45°- 90° Practice
6 Ö2
hypotenuse
¸Ö2
45°
45°
6SAME
6

18. 45°- 45°- 90° Practice
11 Ö2
hypotenuse
¸Ö2
45°
45°

19. 45°- 45°- 90° Practice
11 Ö2
Ö2
= 11

20. 45°- 45°- 90° Practice
11Ö2
hypotenuse
¸Ö2
45°
45°
11SAME
11

21. 45°- 45°- 90° Practice
8
hypotenuse
¸Ö2
45°
45°

22. 45°- 45°- 90° Practice
8
Ö2
Ö2
Ö2
* =
8Ö2
2
= 4Ö2

23. 45°- 45°- 90° Practice
8
hypotenuse
¸Ö2
45°
45°
4Ö2 SAME
4Ö2

24. 45°- 45°- 90° Practice
4
hypotenuse
¸Ö2
45°
45°

25. 45°- 45°- 90° Practice
4
Ö2
Ö2
Ö2
* =
4Ö2
2
= 2Ö2

26. 45°- 45°- 90° Practice
4
hypotenuse
¸Ö2
45°
45°
2Ö2 SAME
2Ö2

27. 45°- 45°- 90° Practice
6
Hypotenuse
¸Ö2
45°
45°

28. 45°- 45°- 90° Practice
6
Ö2
Ö2
Ö2
* =
6Ö2
2
= 3Ö2

29. 45°- 45°- 90° Practice
6
hypotenuse
¸Ö2
45°
45°
3Ö2 SAME
3Ö2

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. 2
2
2
60° 60°
30°- 60°- 90°
The altitude (also
the angle bisector
and median) cuts
the triangle into
two congruent
triangles.
11
30°30°

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

33. 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.

34. 60°
30°
30°- 60°- 90° Practice
1
2 Ö3
A
2
+ B
2
= C
2
A
2
+ 1
2
= 2
2
A
2
+ 1 = 4
A
2
= 3
A = Ö3

35. 30°- 60°- 90°
Conclusion:
the ratio of
the sides in a
30-60-90
triangle is
1- 2 - Ö3
60°
30°
Ö3
1
2

36. 60°
30°
30°- 60°- 90° Practice
4
8
Hypotenuse =
short leg * 2
4Ö3
The key is to find
the length of the
short side.
Long Leg =
short leg *Ö 3

37. 60°
30°
30°- 60°- 90° Practice
5
10
Hypotenuse =
short leg * 2 5Ö3
Long Leg =
short leg *Ö 3

38. 60°
30°
30°- 60°- 90° Practice
7
14
Hypotenuse =
short leg * 2 7Ö3
Long Leg =
short leg *Ö 3

39. 60°
30°
30°- 60°- 90° Practice
Ö3
2Ö3
Hypotenuse =
short leg * 2 3
Long Leg =
short leg *Ö 3

40. 60°
30°
30°- 60°- 90° Practice
Ö10
2Ö10
Hypotenuse =
short leg * 2
Ö30
Long Leg =
short leg *Ö 3

41. 30°- 60°- 90° Practice

42. 60°
30°
30°- 60°- 90° Practice
11
22
Short Leg =
Hypotenuse ¸ 2 11Ö3
Long Leg =
short leg *Ö 3

43. 60°
30°
30°- 60°- 90° Practice
2
4
Short Leg =
Hypotenuse ¸ 2 2Ö3
Long Leg =
short leg *Ö 3

44. 60°
30°
30°- 60°- 90° Practice
9
18
Short Leg =
Hypotenuse ¸ 2 9Ö3
Long Leg =
short leg *Ö 3

45. 60°
30°
30°- 60°- 90° Practice
15
30
Short Leg =
Hypotenuse ¸ 2 15Ö3
Long Leg =
short leg *Ö 3

46. 60°
30°
30°- 60°- 90° Practice
23
46
Hypotenuse =
Short Leg * 2 23Ö3
Short Leg =
Long leg ¸ Ö 3

47. 60°
30°
30°- 60°- 90° Practice
14
28
Hypotenuse =
Short Leg * 2 14Ö3
Short Leg =
Long leg ¸ Ö 3

48. 60°
30°
30°- 60°- 90° Practice
16
32
Hypotenuse =
Short Leg * 2 16Ö3
Short Leg =
Long leg ¸ Ö 3

49. 60°
30°
30°- 60°- 90° Practice
3 Ö3
6Ö3
Hypotenuse =
Short Leg * 2 9
Short Leg =
Long leg ¸ Ö 3

50. 60°
30°
30°- 60°- 90° Practice
4 Ö3
8Ö3
Hypotenuse =
Short Leg * 2 12
Short Leg =
Long leg ¸ Ö 3

51. 60°
30°
30°- 60°- 90° Practice
9 Ö3
18Ö3
Hypotenuse =
Short Leg * 2 27
Short Leg =
Long leg ¸ Ö 3

52. 60°
30°
30°- 60°- 90° Practice
7 Ö3
14Ö3
Hypotenuse =
Short Leg * 2 21
Short Leg =
Long leg ¸ Ö 3

53. 60°
30°
30°- 60°- 90° Practice
11Ö3
22Ö3
Hypotenuse =
Short Leg * 2 33
Short Leg =
Long leg ¸ Ö 3