Question 416·Easy·Math

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In a right triangle, the length of the hypotenuse is 10, and one of the acute angles measures $30^{\circ}$ . What is the length of the side opposite the $30^{\circ}$ angle?

A right triangle with a $30^\circ$ angle is a special right triangle. Its side pattern makes the side opposite $30^\circ$ half as long as the hypotenuse, so identify which side the question asks for before using the given length. The tempting radical answer is the longer leg, across from $60^\circ$ , not the requested side.

Hints

Hint 1
The hypotenuse is across from the right angle. In a right triangle with a $30^\circ$ angle, the side across from that angle has a special relationship to the hypotenuse. Which side does the question ask you to find?
Hint 2
In a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle, the short leg lies opposite $30^\circ$ . The hypotenuse is twice that leg, so write the given hypotenuse as twice the length you need.

Step-by-step

Use the $30^\circ$ special-triangle ratio
Step 1
Identify the side pattern
No Desmos needed. The special-triangle ratio leaves one division. Since the triangle is right and has a $30^\circ$ angle, it’s a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle. Its sides have lengths $x$ , $x\sqrt3$ , and $2x$ . The side opposite $30^\circ$ is $x$ ; the hypotenuse, across from the right angle, is $2x$ .
Step 2
Match the given length to the hypotenuse
The hypotenuse is $10$ , so match $10$ to $2x$ : $2x=10$ . The side opposite $30^\circ$ is half the hypotenuse. Don’t use $x\sqrt3$ here: that’s the leg opposite $60^\circ$ .
Step 3
Find the opposite side
Divide both sides by $2$ to find the short leg:
$x=\frac{10}{2}$
$x=5$ . So the side opposite the $30^\circ$ angle is $5$ . Choice D.

Question 416·Easy·Math

Strategy

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Step-by-step

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Strategy

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