1. Lesson 56: Special Right Triangles
30°- 60°- 90°

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

3. 30°- 60°- 90°
The altitude (also the angle bisector, median, and perpendicular bisector) cuts the triangle into two congruent triangles. 2
60°
30°
30° 1 1

4. 30°- 60°- 90°
60°
30°
This creates the triangle with a hypotenuse a short leg and a long leg.
Long Leg
Hypotenuse
Short Leg

5. 30°- 60°- 90° Practice
We see that the hypotenuse is twice the short leg.
60°
30° 2
We can use the Pythagorean Theorem to find the long leg. 1

6. 2 3 1 30°- 60°- 90° Practice A2 + B2 = C2 A2 + 12 = 22 A2 + 1 = 4

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

8. Properties of a 30°- 60°- 90° Triangles Page 368
In a 30°- 60°- 90° triangle, the length of the hypotenuse is twice the length of the short leg, and the length of the longer leg is the length of the shorter leg times 3 . 2x
x 3 x

9. 30°- 60°- 90° Practice
60°
30°
The key is the length of the short side. 8
4 3
Hypotenuse = short leg * 2 4
Long Leg = short leg * 3

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

11. 14 7 3 7 30°- 60°- 90° Practice Hypotenuse = short leg * 2
7 3 7
Long Leg = short leg * 3

12. 2 3 3 3 30°- 60°- 90° Practice Hypotenuse = short leg * 2
2 3 3
Hypotenuse = short leg * 2 3
Long Leg = short leg * 3

13. 2 10 30 10 30°- 60°- 90° Practice Hypotenuse = short leg * 2
2 10 30
Hypotenuse = short leg * 2 10
Long Leg = short leg * 3

14. 30°- 60°- 90° Practice
Now Let's
Go Backwards

15. 22 11 3 11 30°- 60°- 90° Practice Short Leg = Hypotenuse  2
The key is to find the length of the short side.
60°
30° 22
Short Leg = Hypotenuse  2
11 3 11
Long Leg = short leg * 3

16. 4 2 3 2 30°- 60°- 90° Practice Short Leg = Hypotenuse  2
The key is to find the length of the short side.
60°
30° 4
Short Leg = Hypotenuse  2
2 3 2
Long Leg = short leg * 3

17. 18 9 3 9 30°- 60°- 90° Practice Short Leg = Hypotenuse  2
The key is to find the length of the short side.
60°
30° 18
Short Leg = Hypotenuse  2
9 3 9
Long Leg = short leg * 3

18. 30 15 3 15 30°- 60°- 90° Practice Short Leg = Hypotenuse  2
The key is to find the length of the short side.
60°
30° 30
Short Leg = Hypotenuse  2
15 3 15
Long Leg = short leg * 3

19. 46 23 3 23 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2
The key is to find the length of the short side.
60°
30° 46
Hypotenuse = Short Leg * 2
23 3 23
Short Leg = Long leg  3

20. 28 14 3 14 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2
The key is to find the length of the short side.
60°
30° 28
Hypotenuse = Short Leg * 2
14 3 14
Short Leg = Long leg  3

21. 32 16 3 16 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2
The key is to find the length of the short side.
60°
30° 32
Hypotenuse = Short Leg * 2
16 3 16
Short Leg = Long leg  3

22. 9 3 3 6 3 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2
The key is to find the length of the short side.
60°
30°
6 3
Hypotenuse = Short Leg * 2 9
3 3
Short Leg = Long leg  3

23. 12 4 3 8 3 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2
The key is to find the length of the short side.
60°
30°
8 3
Hypotenuse = Short Leg * 2 12
4 3
Short Leg = Long leg  3

24. 27 9 3 18 3 30°- 60°- 90° Practice Hypotenuse = Short Leg * 2
The key is to find the length of the short side.
60°
30°
18 3
Hypotenuse = Short Leg * 2 27
9 3
Short Leg = Long leg  3

25. 2 30°- 60°- 90° Practice 4 3 3 2 3 3 Hypotenuse = Short Leg * 2
The key is to find the length of the short side.
60°
30°
Hypotenuse = Short Leg * 2 2
Short Leg = Long leg  3

26. 7 30°- 60°- 90° Practice 14 3 3 7 3 3 Hypotenuse = Short Leg * 2
The key is to find the length of the short side.
60°
30°
Hypotenuse = Short Leg * 2 7
Short Leg = Long leg  3

27. THE END