Which answer choice correctly solves for x and y?

Step-by-step explanation:

1. Approach

The easiest method to solve this problem is to use the side ratios in a special right triangle. One should start by proving that the triangle is a (30 - 60 - 90) triangle. Since the problem gives on the information that one of the sides has a measure of (), one can use this combination with the ratio of the sides in a special right triangle, to find the unknown side lengths.

2. Prove this triangle is a (30 - 60 - 90) triangle

One is given a right triangle. This means the triangle has a (90) degree or right angle in it. This is indicated by a box around one of the angles. One is given that the other angle in this triangle has an angle measure of (30) degrees. The problem asks for one to find the third angle measure. A property of any triangle is that the sum of angle measures in the triangle is (180) degrees. One can use this to their advantage by stating the following:

Simplify,

Inverse operations,

Thus, this triangle is a (30 - 60 - 90) triangle, as its angles have the measures of (30 - 60 - 90).

3. Solve for (y)

The sides ratio in a (30 - 60 - 90) triangle is the following:

Where (n) is the side opposite the (30) degree angle, () is the side opposite the (60) degree angle and finally (2n) is the side opposite the (90) degree angle. The side (y) is opposite the (30) degree angle. This means that it is equal to the side opposite the (60) degree angle divided by (). Therefore, one can state the following:

4. Solve for (x)

Using the same thought process one used to solve for side (y), one can solve for side (x). The side (x) is opposite the (90) degree angle, hence, one can conclude that it is twice the length of the side with the length of (y). Therefore, one can state the following: