Question

select the correct answer.
what is the value of x in the triangle?
a. $15\sqrt{3}$
b. 7.5
c. 5
d. $5\sqrt{3}$
e. 15

Explanation:

Step1: Use trigonometric ratio

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here \(\theta = 30^{\circ}\), the adjacent side is \(15\) and the opposite side is \(x\). But we can also use the property of \(30 - 60-90\) triangle. In a \(30 - 60 - 90\) triangle, if the side adjacent to \(30^{\circ}\) (longer leg) is \(a\), and the side opposite to \(30^{\circ}\) (shorter leg) is \(b\), the relationship is \(b=\frac{a}{\sqrt{3}}\). Another way: \(\tan30^{\circ}=\frac{x}{15}\).
Since \(\tan30^{\circ}=\frac{1}{\sqrt{3}}\), we have the equation \(\frac{1}{\sqrt{3}}=\frac{x}{15}\).

Step2: Solve for \(x\)

Cross - multiply the equation \(\frac{1}{\sqrt{3}}=\frac{x}{15}\) gives \(x=\frac{15}{\sqrt{3}}\). Rationalize the denominator: \(x = \frac{15\sqrt{3}}{3}=5\sqrt{3}\).

Answer:

D. \(5\sqrt{3}\)