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Lesson 56:Special Right Triangles
30°- 60°- 90°
30°- 60°- 90°
The 30-60-90 triangle is based on an equilateral triangle with sides of 2 units.
The 30-60-90 triangle is based on an equilateral triangle with sides of 2 units.
22
2
60° 60°
22
2
60° 60°
30°- 60°- 90°
The altitude (also the angle bisector, median, and perpendicular bisector) cuts the triangle into two congruent triangles.
The altitude (also the angle bisector, median, and perpendicular bisector) cuts the triangle into two congruent triangles.
11
30°30°
This creates the 30-60-90 triangle with a hypotenuse a short leg and a long leg.
This creates the 30-60-90 triangle with a hypotenuse a short leg and a long leg.
30°- 60°- 90°
Hypotenuse
HypotenuseShort LegShort Leg
Long
Leg
Long
Leg
60°
30°
60°
30°
30°- 60°- 90° Practice
1
2
We see that the hypotenuse is twice the short leg.
We see that the hypotenuse is twice the short leg.
We can use the Pythagorean Theorem to find the long leg.
We can use the Pythagorean Theorem to find the long leg.
60°
30°
30°- 60°- 90° Practice
1
2 √3
A2 + B2 = C2
A2 + 12 = 22
A2 + 1 = 4
A2 = 3
A =
A2 + B2 = C2
A2 + 12 = 22
A2 + 1 = 4
A2 = 3
A =
30°- 60°- 90°
Conclusion: the ratio of the sides in a 30-60-90 triangle is 1- -2
Conclusion: the ratio of the sides in a 30-60-90 triangle is 1- -2
60°
30°
√3
1
2
Properties of a 30°- 60°- 90° TrianglesPage 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 .
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 .
60°
30°
x
x
2x
60°
30°
30°- 60°- 90° Practice
4
8
Hypotenuse = short leg * 2
4
The key is the length of the short side.
The key is the length of the short side.
Long Leg = short leg *
60°
30°
30°- 60°- 90° Practice
5
10 Hypotenuse = short leg * 2 5
Long Leg = short leg *
60°
30°
30°- 60°- 90° Practice
7
14 Hypotenuse = short leg * 2 7
Long Leg = short leg *
60°
30°
30°- 60°- 90° Practice
√3
2 Hypotenuse = short leg * 2 3
Long Leg = short leg *
60°
30°
30°- 60°- 90° Practice
√10
2Hypotenuse = short leg * 2
√30
Long Leg = short leg *
30°- 60°- 90° Practice
60°
30°
30°- 60°- 90° Practice
11
22 Short Leg = Hypotenuse 2 11
Long Leg = short leg *
The key is to find the length of the short side.
60°
30°
30°- 60°- 90° Practice
2
4 Short Leg = Hypotenuse 2 2
Long Leg = short leg *
The key is to find the length of the short side.
60°
30°
30°- 60°- 90° Practice
9
18 Short Leg = Hypotenuse 2 9
Long Leg = short leg *
The key is to find the length of the short side.
60°
30°
30°- 60°- 90° Practice
15
30 Short Leg = Hypotenuse 2 15
Long Leg = short leg *
The key is to find the length of the short side.
60°
30°
30°- 60°- 90° Practice
23
46 Hypotenuse = Short Leg * 2 23
Short Leg = Long leg
The key is to find the length of the short side.
60°
30°
30°- 60°- 90° Practice
14
28 Hypotenuse = Short Leg * 2 14
Short Leg = Long leg
The key is to find the length of the short side.
60°
30°
30°- 60°- 90° Practice
16
32 Hypotenuse = Short Leg * 2 16
Short Leg = Long leg
The key is to find the length of the short side.
60°
30°
30°- 60°- 90° Practice
3
6Hypotenuse = Short Leg * 2 9
Short Leg = Long leg
The key is to find the length of the short side.
60°
30°
30°- 60°- 90° Practice
4
8 Hypotenuse = Short Leg * 2 12
Short Leg = Long leg
The key is to find the length of the short side.
60°
30°
30°- 60°- 90° Practice
9
18Hypotenuse = Short Leg * 2 27
Short Leg = Long leg
The key is to find the length of the short side.
60°
30°
30°- 60°- 90° Practice
2√33
4 √33 Hypotenuse =
Short Leg * 2 2
Short Leg = Long leg
The key is to find the length of the short side.
60°
30°
30°- 60°- 90° Practice
7√33
14√33 Hypotenuse =
Short Leg * 2 7
Short Leg = Long leg
The key is to find the length of the short side.
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