In right triangle , the measure of is and the measure of is . If the length of side (opposite ) is , what is the length of side , the hypotenuse of the triangle?

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SAT Lines, Angles & Triangles Question 38 | Free SAT Prep | Aniko

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Question 38·Medium·Lines, Angles, and Triangles

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In right triangle $XYZ$ , the measure of $\angle X$ is $30^\circ$ and the measure of $\angle Y$ is $60^\circ$ . If the length of side $YZ$ (opposite $\angle X$ ) is $5$ , what is the length of side $XY$ , the hypotenuse of the triangle?

For right-triangle questions that mention angles of $30^\circ$ and $60^\circ$ , immediately recognize a 30-60-90 special right triangle and recall the fixed side ratio $1 : \sqrt{3} : 2$ (shortest side : longer leg : hypotenuse). Identify which given side corresponds to which angle so you know whether to multiply the shortest side by 2 (for the hypotenuse) or by $\sqrt{3}$ (for the longer leg). This avoids time-consuming trigonometry and helps you quickly eliminate trap answers based on the 45-45-90 pattern or on mixing up which side gets the $\sqrt{3}$ factor.

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Recognize the triangle type

You are given a right triangle with angles $30^\circ$ and $60^\circ$ . What special name is given to a right triangle with angles $30^\circ$ , $60^\circ$ , and $90^\circ$ ?

Recall the side-length ratio

In a 30-60-90 triangle, the three sides (shortest side, longer leg, and hypotenuse) are always in the same ratio. Try to remember or write down that standard ratio.

Connect the given side to the hypotenuse

Side $YZ$ is opposite $\angle X$ , which is $30^\circ$ . In the 30-60-90 ratio, how does the side opposite $30^\circ$ compare to the hypotenuse?

Use a simple multiplication

Once you know how the hypotenuse relates to the side opposite $30^\circ$ , use the given length 5 to set up a one-step multiplication to find $XY$ .

Desmos Guide

Use Desmos to perform the final multiplication

In a 30-60-90 triangle, once you know that the hypotenuse is twice the side opposite $30^\circ$ , you need to calculate $2 \cdot 5$ . In a Desmos expression line, type 2*5 and look at the numerical result; that value is the length of $XY$ , the hypotenuse.

Step-by-step Explanation

Identify the special right triangle

Because triangle $XYZ$ is a right triangle and $\angle X = 30^\circ$ and $\angle Y = 60^\circ$ , it is a 30-60-90 triangle.

In any 30-60-90 triangle, the side lengths are in the fixed ratio:

\text{opposite }30^\circ : \text{opposite }60^\circ : \text{hypotenuse} = 1 : \sqrt{3} : 2

The side opposite the $30^\circ$ angle is the shortest side, and the hypotenuse is twice that shortest side.

Match the given side to the ratio

We are told that side $YZ$ is opposite $\angle X$ , and $\angle X = 30^\circ$ , so $YZ$ is the side opposite $30^\circ$ .

From the 30-60-90 ratio, if the side opposite $30^\circ$ is some length $s$ , then the hypotenuse is $2s$ .

Here, $s = YZ = 5$ , so the hypotenuse $XY$ must satisfy

XY = 2 \cdot 5

Do not simplify yet; just set up the multiplication.

Compute the hypotenuse

Now simplify the expression for $XY$ :

XY = 2 \cdot 5 = 10

So the length of side $XY$ , the hypotenuse of the triangle, is $10$ . This corresponds to answer choice C.

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Step-by-step Explanation

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Step-by-step Explanation