Answer
$C$
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 measures $60^{\circ}$.
In this triangle, the longer leg is $\sqrt 3$ times the length of the shorter leg. Let's set up that equation to solve for $x$, the length of the shorter leg:
$6 = \sqrt 3(x)$
Divide both sides by $\sqrt 3$ to solve for $x$:
$x = \frac{6}{\sqrt 3}$
To simplify the fraction, we need to get rid of the radical in the denominator by multiplying both the numerator and denominator by the radical:
$x = \frac{6}{\sqrt 3} • \frac{\sqrt 3}{\sqrt 3}$
Multiply:
$x = \frac{6\sqrt 3}{\sqrt 9}$
Take the square root of the denominator:
$x = \frac{6\sqrt 3}{3}$
Divide both the numerator and denominator by their greatest common factor to simplify:
$x = 2 \sqrt 3$
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:
$y = 2(2 \sqrt 3)$
Multiply to solve for $y$:
$y = 4\sqrt 3$
The answer is option $C$.
