Question

10 multiple choice 1 point
find the value of the variable(s). if your answer is not an integer, leave it insimplest radical form.
not drawn to scale
$x = 12\sqrt {3}, y = 24$
$x = 24\sqrt {3}, y = 12$
$x = 12, y = 24\sqrt {3}$
$x = 24, y = 12\sqrt {3}$

Explanation:

Step1: Use trigonometric ratios

In a right - triangle with an angle of \(30^{\circ}\), we know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
For \(\sin30^{\circ}=\frac{12}{y}\), since \(\sin30^{\circ}=\frac{1}{2}\), we have \(\frac{1}{2}=\frac{12}{y}\).
Cross - multiply: \(y = 24\).

Step2: Use \(\tan\) ratio

For \(\tan30^{\circ}=\frac{12}{x}\), and \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\).
So, \(\frac{1}{\sqrt{3}}=\frac{12}{x}\), cross - multiply gives \(x = 12\sqrt{3}\).

Answer:

\(x = 12\sqrt{3},y = 24\) (the first option)