# How Do You Apply the 30-60-90 Triangle Theorem?

The lengths of the sides of a 30-60-90 triangle always exist in the proportional pattern 1:2:sqrt 3. The shorter side is half as long as the hypotenuse, and the length of the longer side is found by multiplying the length of the shorter side by the square root of three. Knowing this pattern makes it possible to find the lengths of the other two sides if one side is given.

1. Use the length of the hypotenuse to find the lengths of the other two sides

As an example, suppose the length of the hypotenuse was 14 inches. The length of the shortest side equals 1/2 the hypotenuse, so that would be 7 inches long. The length of the longer side equals the length of the shorter side times the square root of three; here, 7(sqrt 3) = 12.12 inches, approximately.

2. Use the length of the shortest side to find the lengths of the other two sides

Assume the shorter side is 6 inches. The hypotenuse is twice as long as the shorter leg, so the hypotenuse is 12 inches. The length of the longer leg equals (sqrt 3) times the shortest side, and 6(sqrt 3) = 10.39 inches approximately.

3. Use the length of the longer non-hypotenuse side to find the lengths of the other two sides

Assume the length of the longer side is 8 inches. The length of this side is 1/2(sqrt 3) times the hypotenuse, therefore 8 = 1/2 hypotenuse(sqrt 3). Multiply both sides by 2 to get 16 = hypotenuse(sqrt 3). Divide both sides by (sqrt 3) to get hypotenuse = 16/(sqrt 3), which is approximately 9.24 inches. Find the shorter side by halving the hypotenuse, in this case approximately 4.62 inches.

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