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**Lesson 56: Special Right Triangles**

30°- 60°- 90°

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

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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

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

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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

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

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**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

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**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

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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

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

5 3 5 Long Leg = short leg * 3

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**14 7 3 7 30°- 60°- 90° Practice Hypotenuse = short leg * 2**

7 3 7 Long Leg = short leg * 3

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**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

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**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

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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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THE END

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