Answer
$x = 4\sqrt {3}$
$y = 8\sqrt {3}$
Work Step by Step
The diagram is that of a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle because one angle measures $30^{\circ}$, another measures $90^{\circ}$, and the last angle must measure $60^{\circ}$.
In this triangle, the longer leg is $\sqrt 3$ times the length of the shorter leg. We can set up an equation to solve for $x$, the length of the shorter leg:
$12 = \sqrt 3(x)$
Divide both sides by $\sqrt {3}$ to solve for $x$:
$x = \frac{12}{\sqrt {3}}$
To simplify the fraction, we multiply both the numerator and denominator by the denominator to get rid of the radical in the denominator:
$x = \frac{12}{\sqrt {3}} • \frac{\sqrt {3}}{\sqrt {3}}$
Multiply to simplify:
$x = \frac{12\sqrt {3}}{\sqrt {9}}$
Simplify the denominator:
$x = \frac{12\sqrt {3}}{3}$
Divide both the numerator and denominator by $3$ to simplify the fraction:
$x = 4\sqrt {3}$
In this type of right triangle, the hypotenuse is two times the shorter leg. Let's write an equation to solve for $y$, the length of the hypotenuse:
$y = 2(4\sqrt {3})$
Multiply to solve for $y$:
$y = 8\sqrt {3}$
