special right triangles 30:60:90 right triangles
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Special Right Triangles
30:60:90
Right Triangles
30:60:90 Relationship
Given: Equilateral Triangle with side=2, find the altitude.
222 cba
2 2
21
x
222 21 b41 2 b1 1
3 2 b3b
360º
30º
30:60:90 Relationship
Given: Equilateral Triangle with side=4, find the altitude.
222 cba
4 4
42
x
222 42 b164 2 b
4 4 21 2 b34 b
32
32b
60º
30º
30:60:90 Relationship
Given: Equilateral Triangle with side=10, find the altitude.
222 cba
10 10
105
x
222 105 b10025 2 b
52 25 57 2 b
325 b
35
35b
60º
30º
Conclusion
3
xxx 2:3:
21 3
60º
30º
1
- The side opposite the 30º angle is half the hypotenuse
- The side opposite the 60º angle is half the hypotenuse times
- The ratio of the sides of a 30:60:90 right triangle is
3
Remember, the 30-60-90 triangle always has the same ratio for its sides:
2:3:1
90
60
30
Remember the relationship the sides have with the angles! The smallest side is across from the smallest angle!
7.13
Since 30 is the smallest angle, then the 1 goes across from it!
1Since 60 is the next biggest angle, then the goes across from it!
3 Since 90 is the largest angle, then the 2 goes across from it!
2
Page 25
Page 26
30 :60 : 901 :√3 :25 :
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.
5√3 :10
𝑦=5√3𝑥=10
Page 26
30 :60 : 901 :√3 :2
:2010 :10√3
𝑦=10√3𝑥=10
Multiply by 10
Page 26
30 :60 : 901 :√3 :2:√3 :1 2
𝑦=2𝑥=1
Multiply by 1
Page 26
30 :60 : 901 :√3 :2
:3015 :15√3
𝑦=15√3𝑥=15
Multiply by 15
Page 26
30 :60 : 901 :√3 :2
:30 :
𝑥 ∙√3=30
𝑥 ∙ √3√3
= 30√3
𝑥=30
√3
30
√360
√3
𝑥=30
√3𝑦=
60
√3
If you see this on the Regents and is a multiple choice question, compare decimals to the answer given.
𝑥=30
√3=17.32050808
2 ∙30
√3=
60
√3
Page 26
10 10
10
60 6090
30
5
𝑥
30 :60 : 901 :√3 :2
5 ::105√3Multiply by 5
Page 26
16 16
6
60 6090
30
8
𝑥
30 :60 : 901 :√3 :2
8 ::168 √3Multiply by 8
Page 26
4 4
4
60 6090
30
2
𝑥
30 :60 : 901 :√3 :2
2 :: 42√3Multiply by 2
Page 26
7 7
7
60 6090
30
3.5
𝑥
30 :60 : 901 :√3 :2
3 .5 : :73.5√3Multiply by 3.5
Page 26
15 15
15
60 6090
30
7 .5
𝑥
30 :60 : 901 :√3 :2
7.5 : :157.5√3Multiply by 7.5
Page 26
𝑠 𝑠
𝑠
60 6090
30
𝑠2
𝑥
30 :60 : 901 :√3 :2
𝑠2
:: 𝑠𝑠2
√3
Multiply by
Page 26
60 6090
30
7√3
30 :60 : 901 :√3 :2:7√3 :7 14
7
14
Multiply by 7
Page 26
10
14
30 90
60
30 :60 : 901 :√3 :2
:105 :5√3Multiply by 5
5
Page 26
90
60
30
8
30 :60 : 901 :√3 :2
:8Multiply by 4
4 : 4√3
4√34√3=6.92=6.9
Page 26
603030
90
6010 30 90
6010
30 :60 : 901 :√3 :2
:10Multiply by 5
5 :5√3
5
5
5
h𝑆 𝑜𝑟𝑡𝑒𝑟 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑖𝑠5+5=10
𝐿𝑜𝑛𝑔𝑒𝑟 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙𝑖𝑠5 √3+5√3=10√3
Page 26
30 :60 : 901 :√3 :2
:8 √3
𝑥 ∙2=8√3
𝑥 ∙ 22=8 √3
2
𝑥=4√3
: √3 ∙4 √3=4 √9¿ 4 ∙3¿12
12
4√312
30 :60 : 901 :√3 :2: 4√3 :
Multiply by 4
4 8
4
8
𝐴𝐵=4+12=16
Homework
Page 26#12,14,16,19
Separate Sheet
Page 26
A B
C
1208
60
9030
30 :60 : 901 :√3 :2
:84 : 4√3Multiply by 4
4√3
4
4√3
4√3+4 √3=13.8¿14
Page 26
30
120
180−120=60602
=3030
3060
30 :60 : 901 :√3 :2
5 :5√3 :10Multiply by 5
10 5√3
10
30 :60 : 901 :√3 :2
5√3 :15 :Multiply by
10√3
10√3
15
Page 26
3060
6
30 :60 : 901 :√3 :2
:63 :3 √3Multiply by 3
3
3√3
30 :60 : 901 :√3 :26 :
Multiply by 66 √3 :12
12
6 √3
Page 26
6 √3
30 :60 : 901 :√3 :2:6 √3 :6 12
Multiply by 6
6
12
30 :60 : 901 :√3 :2
12 :12√3 :24Multiply by 1224
24−6=18
12√3
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