Question

find a. the right - triangle has a right angle, a 60° angle, one leg is (12sqrt{3}), the other leg is (a), and the hypotenuse is (b). the options are (24sqrt{3}), (12), (24), (16).

Explanation:

Step1: Identify trigonometric ratio

We use the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta=60^\circ$, opposite side is $12\sqrt{3}$, adjacent side is $a$.

Step2: Substitute values into formula

$\tan(60^\circ) = \frac{12\sqrt{3}}{a}$
We know $\tan(60^\circ)=\sqrt{3}$, so:
$\sqrt{3} = \frac{12\sqrt{3}}{a}$

Step3: Solve for $a$

Rearrange to isolate $a$:
$a = \frac{12\sqrt{3}}{\sqrt{3}}$
Cancel $\sqrt{3}$ in numerator and denominator:
$a=12$

Answer:

B. 12