Special Right Triangles
30:60:90
Right Triangles
30:60:90 Relationship

Given: Equilateral
Triangle with side=2,
find the altitude.
30º
2
2
x3
a b
2
2
1 b
2
1 b
1
2
2
b
60º
1
2
2
c
2
2
4
1
2

b
3
3
30:60:90 Relationship

Given: Equilateral
Triangle with side=4,
find the altitude.
30º
4
4
2x 3
a b c
2
2
2
2 b  4
2
4  b  16
4
4
2
b
60º
2
4
2
2
b

2
12
4 3
b2 3
30:60:90 Relationship

Given: Equilateral
Triangle with side=10,
find the altitude.
30º
a b c
2
5  b  10
2
2
2
2
2
25  b  100
 25
 25
2
10
10
5x 3
b
60º
5
10
2
b

75
25  3
b5 3
Conclusion
- The side opposite the 30º
angle is half the hypotenuse
30º
2
1
60º
1
3
3
- The side opposite the 60º
angle is half the hypotenuse
times 3
- The ratio of the sides of a
30:60:90 right triangle is
x : x 3 : 2x
Page 25
Remember, the 30-60-90 triangle always has the same ratio for its sides:
1:
3 :2
Remember the relationship the
sides have with the angles! The
smallest side is across from the
smallest angle!
3  1 .7
60 
2
Since 30 is the smallest angle,
then the 1 goes across from it!
1
90 
30 
3
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!
Page 26
30: 60: 90
1: 3: 2
5: 5 3: 10
𝑦=5 3
𝑥 = 10
Now, since the ratio
is always the same,
then what did we
multiply by?
Five! If we multiply
one number in the
ratio by 5, we
multiply all of them
by 5.
Page 26
30: 60: 90
1: 3: 2
10: 10 3 : 20
𝑦 = 10 3
𝑥 = 10
Multiply by 10
Page 26
30: 60: 90
1: 3: 2
1: 3: 2
𝑥=1
𝑦=2
Multiply by 1
Page 26
30: 60: 90
1: 3: 2
15: 15 3 : 30
𝑥 = 15
𝑦 = 15 3
Multiply by 15
Page 26
30: 60: 90
1: 3: 2
30
60
: 30:
3
3
𝑥=
30
3
2∙
30
𝑦=
If you see this on the Regents
and is a multiple choice
question, compare decimals to
the answer given.
3
=
60
3
60
3
𝑥=
30
3
𝑥 ∙ 3 = 30
3 30
𝑥∙
=
3
3
30
𝑥=
3
= 17.32050808
Page 26
30: 60: 90
1: 3: 2
30
10
60
5: 5 3 ∶ 10
10
𝑥
90 5
10
60
Multiply by 5
Page 26
30: 60: 90
1: 3: 2
30
16
60
8: 8 3 ∶ 16
16
𝑥
90 8
16
60
Multiply by 8
Page 26
30: 60: 90
1: 3: 2
30
4
4
𝑥
90 2
60
4
2: 2 3 ∶ 4
60
Multiply by 2
Page 26
30: 60: 90
1: 3: 2
30
7
𝑥
3.5: 3.5 3 ∶ 7
7
90 3.5 60
60
7
Multiply by 3.5
Page 26
30: 60: 90
1: 3: 2
30
15
60
7.5: 7.5 3 ∶ 15
15
𝑥
90 7.5 60
15
Multiply by 7.5
Page 26
30: 60: 90
30
𝑠
𝑠
𝑥
90
60
𝑠
𝑠
2 60
1: 3: 2
𝑠 𝑠
3∶𝑠
:
2 2
𝑠
Multiply by 2
Page 26
30: 60: 90
1: 3: 2
30
7 : 7 3: 14
14
7 3
60
90 7
60
Multiply by 7
Page 26
30: 60: 90
60
10
5
90
30
14
1: 3: 2
5: 5 3 : 10
Multiply by 5
Page 26
30: 60: 90
1: 3: 2
60
4: 4 3 : 8
8
90
30
4 3
4 3 = 6.92 = 6.9
Multiply by 4
Page 26
10
60 5
90
5 3
30
6030
10
5 3
5
𝑆ℎ𝑜𝑟𝑡𝑒𝑟 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑖𝑠 5 + 5 = 10
𝐿𝑜𝑛𝑔𝑒𝑟 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑖𝑠 5 3 + 5 3 = 10 3
30
5 3
60
5
90
30: 60: 90
1: 3: 2
5: 5 3 : 10
Multiply by 5
Page 26
8 3
4 3
8
12
𝑥∙2=8 3
2 8 3
𝑥∙ =
2
2
4
30: 60: 90
1: 3: 2
4 3: 12 : 8 3
30: 60: 90
1: 3: 2
4 : 4 3: 8
𝐴𝐵 = 4 + 12 = 16
Multiply by 4
𝑥=4 3
3∙4 3=4 9
=4∙3
= 12
Homework
Page
26
#12,14,16,19
Separate Sheet
Page 26
C
30: 60: 90
120
60
8
4
90
30
A
1: 3: 2
4: 4 3 : 8
4 3
4 3
4 3 + 4 3 = 13.8
= 14
B
Multiply by 4
180 − 120 = 60
60
= 30
2
Page 26
30 30
10 3
5 3
120
30
60
10
10
15
30: 60: 90
1: 3: 2
5: 5 3: 10
Multiply by 5
30: 60: 90
1: 3: 2
5 3:15: 10 3
Multiply by 5 3
Page 26
60
6 3
30 6
3 3
30: 60: 90
1: 3: 2
3: 3 3 : 6
3
Multiply by 3
30: 60: 90
1: 3: 2
6: 6 3: 12
12
Multiply by 6
Page 26
30: 60: 90
12 3
12
6 3
1: 3: 2
Multiply by 6
6 : 6 3: 12
24 − 6 = 18
24
6
30: 60: 90
1: 3: 2
12:12 3: 24
Multiply by 12