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Difficulty: EasyPythagorean Theorem and Special Right Triangles

In a right triangle, the length of the side opposite the 60∘60^\circ angle is 636\sqrt{3} centimeters. What is the length, in centimeters, of the hypotenuse of this triangle?

Answer: 12 cm

Answer

The length of the hypotenuse is 1212 centimeters.
In a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ special right triangle, the sides opposite the 30∘30^\circ angle, the 60∘60^\circ angle, and the 90∘90^\circ (hypotenuse) angle are in the ratio x:x3:2xx : x\sqrt{3} : 2x. Given that the side opposite the 60∘60^\circ angle is 636\sqrt{3} centimeters, we have x3=63x\sqrt{3} = 6\sqrt{3}, which means x=6x = 6. The hypotenuse is 2x=2(6)=122x = 2(6) = 12 centimeters.

Step-by-Step Solution

1
Determine the type of special right triangle.
A 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle.
Since the triangle is a right triangle and has a 60∘60^\circ angle, the remaining angle must be 180∘−90∘−60∘=30∘180^\circ - 90^\circ - 60^\circ = 30^\circ.
2
Set up the relation for the side lengths using the ratio of a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle.
The side opposite the 60∘60^\circ angle is x3x\sqrt{3} centimeters, where xx is the length of the side opposite the 30∘30^\circ angle.
In any 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle, the side lengths are in the ratio 1:3:21 : \sqrt{3} : 2.
3
Solve for the base variable xx.
x=6x = 6
We are given that the side opposite the 60∘60^\circ angle is 636\sqrt{3} centimeters, so x3=63x\sqrt{3} = 6\sqrt{3}.
4
Calculate the length of the hypotenuse.
The hypotenuse is 2x=2(6)=122x = 2(6) = 12 centimeters.
The hypotenuse of a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle is twice the length of the shorter leg, which is 2x2x.

Key Concept

Using the side length ratios of a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ special right triangle to find missing lengths.
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Topics

Pythagorean Theorem and Special Right Triangles

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