1. Special Right Triangles
30:60:90
Right Triangles

2. 30:60:90 Relationship
Given: Equilateral Triangle with side=2, find the altitude.
30º 2 2 x
60º 1 2

3. 30:60:90 Relationship
Given: Equilateral Triangle with side=4, find the altitude.
30º 4 4 x
60º 2 4

4. 30:60:90 Relationship
Given: Equilateral Triangle with side=10, find the altitude.
30º 10 10 x
60º 5 10

5. Conclusion - The side opposite the 30º angle is half the hypotenuse
- The side opposite the 60º angle is half the hypotenuse times 2 1
- The ratio of the sides of a 30:60:90 right triangle is
60º 1

6. Since 30 is the smallest angle, then the 1 goes across from it!
Page 25
Remember, the triangle always has the same ratio for its sides:
Remember the relationship the sides have with the angles! The smallest side is across from the smallest angle!
Since 30 is the smallest angle, then the 1 goes across from it!
Since 60 is the next biggest angle, then the 3 goes across from it!
Since 90 is the largest angle, then the 2 goes across from it!

7. Page 26
30:60:90
Now, since the ratio is always the same, then what did we multiply by?
1: 3 :2 5:
5 3 :10
Five! If we multiply one number in the ratio by 5, we multiply all of them by 5.
𝑦=5 3
𝑥=10

8. Page 26
30:60:90
1: 3 :2
Multiply by 10
10:10 3
:20
𝑦=10 3
𝑥=10

9. Page 26
30:60:90
1: 3 :2
Multiply by 1
: 3 :
𝑥=1
𝑦=2

10. Page 26
30:60:90
1: 3 :2
Multiply by 15
15:15 3
:30
𝑥=15
𝑦=15 3

11. Page 26
30:60:90
2∙ =
𝑥∙ 3 =30
1: 3 :2
𝑥∙ =
:30: 𝑥= 𝑥= 𝑦=
If you see this on the Regents and is a multiple choice question, compare decimals to the answer given.
𝑥= =

12. Page 26
30:60:90
1: 3 :2
Multiply by 5 30
5: :10
5 3 10 10 𝑥 60 90 5 60 10

13. Page 26
30:60:90
1: 3 :2
Multiply by 8 30
8: :16
8 3 16 16 𝑥 60 90 8 60 16

14. Page 26
30:60:90
1: 3 :2
Multiply by 2 30
2: :4
2 3 4 4 𝑥 60 90 2 60 4

15. Page 26
30:60:90
1: 3 :2
Multiply by 3.5 30
3.5: :7
3.5 3 7 7 𝑥 60 90
3.5 60 7

16. Page 26
30:60:90
1: 3 :2
Multiply by 7.5 30
7.5: :15
7.5 3 15 15 𝑥 60 90
7.5 60 15

17. Page 26
30:60:90
1: 3 :2
Multiply by 𝑠 2 30
𝑠 2 : :𝑠
𝑠 2 3 𝑠 𝑠 𝑥
𝑠 2 60 90 60 𝑠

18. Page 26
30:60:90
1: 3 :2
Multiply by 7 30
:7 3 : 14
7 3 60 90 7 60

19. Page 26
30:60:90
1: 3 :2
Multiply by 5 60 10
5:5 3
:10 5 30 90 14

20. Page 26
30:60:90
1: 3 :2
Multiply by 4 60
4:4 3 :8 8 90 30
4 3
4 3 =6.92=6.9

21. Page 26 60 10 60 5 5
5 3 10 30 90 90
5 3
5 3 5 30 60 30
30:60:90
1: 3 :2
𝑆ℎ𝑜𝑟𝑡𝑒𝑟 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑖𝑠 5+5=10
Multiply by 5
5:5 3
:10
𝐿𝑜𝑛𝑔𝑒𝑟 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑖𝑠 =10 3

22. Page 26
8 3 8
4 3 12
𝑥∙2=8 3 4
𝑥∙ 2 2 =
30:60:90
30:60:90
𝑥=4 3
Multiply by 4
1: 3 :2
1: 3 :2
4 3 : 12
:8 3
:4 3 :
3 ∙4 3 =4 9
=4∙3
=12
𝐴𝐵=4+12=16

23. Homework
Page 26
#12,14,16,19
Separate Sheet

24. Page 26 A B C
30:60:90 60
120
1: 3 :2 8
Multiply by 4
4:4 3 :8 4 30 90
4 3
4 3
=13.8
=14

25. 180−120=60
Page 26
60 2 =30 30 30
10 3 10
5 3 30
120 60 10
30:60:90 15
1: 3 :2
Multiply by 5
30:60:90 5:
5 3 :10
1: 3 :2
Multiply by 5 3
5 3 :
15:
10 3

26. Page 26 60
6 3 30 6
3 3
30:60:90 3
1: 3 :2
Multiply by 3
3:3 3 :6 12
30:60:90
1: 3 :2
Multiply by 6 6:
6 3 :12

27. Page 26
30:60:90
12 3
1: 3 :2 12
6 3
Multiply by 6
:6 3 : 6
24−6=18
30:60:90
1: 3 :2
Multiply by 12 24
12:
12 3 :24