Question

find the value of each variable. if your answer is not an integer, write it in simplest radical form with the denominator rationalized.
(simplify your answers.)
x = \square and y = \square

Explanation:

Step1: Identify triangle type

This is a 30-60-90 right triangle, where sides follow the ratio $1:\sqrt{3}:2$ (short leg : long leg : hypotenuse).

Step2: Match given side to ratio

The side $12\sqrt{3}$ is opposite the $30^\circ$ angle, so it is the short leg.

Step3: Solve for hypotenuse $x$

Hypotenuse is twice the short leg.
$x = 2 \times 12\sqrt{3} = 24\sqrt{3}$

Step4: Solve for long leg $y$

Long leg is short leg times $\sqrt{3}$.
$y = 12\sqrt{3} \times \sqrt{3} = 12 \times 3 = 36$

Answer:

$x = 24\sqrt{3}$ and $y = 36$