Question

in the right triangle shown, $m\angle a = 30^{\circ}$ and $bc = 6\sqrt{2}$. how long is $ac$? choose 1 answer: a $3\sqrt{6}$ b 9 c 12 d $6\sqrt{6}$ e $12\sqrt{2}$

Explanation:

Step1: Identify trigonometric ratio

In right $\triangle ABC$, $\angle C=90^\circ$, $\angle A=30^\circ$. We use $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, so $\tan(30^\circ)=\frac{BC}{AC}$.

Step2: Substitute known values

We know $BC=6\sqrt{2}$, $\tan(30^\circ)=\frac{1}{\sqrt{3}}$. Substitute into the ratio:
$\frac{1}{\sqrt{3}}=\frac{6\sqrt{2}}{x}$ where $x=AC$.

Step3: Solve for $x$

Rearrange to solve for $x$:
$x=6\sqrt{2}\times\sqrt{3}=6\sqrt{6}$

Answer:

D. $6\sqrt{6}$