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Question

Difficulty: EasyPythagorean Theorem and Special Right Triangles

In a right triangle, the measure of one of the acute angles is 30∘30^\circ. If the side opposite this 30∘30^\circ angle has a length of 6.56.5 inches, what is the length, in inches, of the hypotenuse?

Answer: 13 inches

Answer

The length of the hypotenuse is 1313 inches.
In a 30∘30^\circ-60∘60^\circ-90∘90^\circ special right triangle, the length of the hypotenuse is exactly twice the length of the shorter leg (the side opposite the 30∘30^\circ angle). Given that the shorter leg has a length of 6.56.5 inches, the hypotenuse has a length of 2×6.5=132 \times 6.5 = 13 inches.

Step-by-Step Solution

1
Determine the relationship between the given side and the hypotenuse using special right triangle properties.
The triangle is a 30∘30^\circ-60∘60^\circ-90∘90^\circ right triangle, meaning the hypotenuse is twice the length of the shorter leg.
By geometric theorem, the sides of a 30∘30^\circ-60∘60^\circ-90∘90^\circ triangle are in the ratio 1:3:21 : \sqrt{3} : 2, with the shortest side opposite the 30∘30^\circ angle and the longest side being the hypotenuse.
2
Multiply the length of the side opposite the 30∘30^\circ angle by 2.
6.5 inches×2=13 inches6.5 \text{ inches} \times 2 = 13 \text{ inches}
Doubling the shorter leg length of 6.56.5 inches gives the length of the hypotenuse.

Key Concept

In a 30∘30^\circ-60∘60^\circ-90∘90^\circ special right triangle, the length of the hypotenuse is always twice the length of the shorter leg (the side opposite the 30∘30^\circ angle).
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Topics

Pythagorean Theorem and Special Right Triangles

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