Question

select the correct answer.
in right triangle xyz, ∠y is the right angle and m∠x = 60°. if yz = 4, what is xy?
○ a. 4
○ b. 4√3/3
○ c. 4√3
○ d. 8

Explanation:

Step1: Recall the tangent function

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). In right - triangle \(XYZ\) with \(\angle Y = 90^{\circ}\), \(\angle X=60^{\circ}\), the side opposite to \(\angle X\) is \(YZ\) and the side adjacent to \(\angle X\) is \(XY\). So, \(\tan X=\frac{YZ}{XY}\).

Step2: Substitute the known values

Since \(\tan60^{\circ}=\sqrt{3}\) and \(YZ = 4\), we have \(\sqrt{3}=\frac{4}{XY}\).

Step3: Solve for \(XY\)

Cross - multiply to get \(XY\times\sqrt{3}=4\), then \(XY=\frac{4}{\sqrt{3}}\). Rationalize the denominator: \(XY=\frac{4\sqrt{3}}{3}\).

Answer:

B. \(\frac{4\sqrt{3}}{3}\)