Question

given right triangle gyk, what is the value of tan(g)?
options:
$\frac{1}{2}$
$\frac{sqrt{3}}{2}$
$\frac{2sqrt{3}}{3}$
$sqrt{3}$
(image shows right triangle gyk with right angle at k, angle at g is 60°, angle at y is 30°, and gk = 27)

Explanation:

Step1: Recall the tangent formula

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(\angle G\), the opposite side is \(YK\) and the adjacent side is \(GK = 27\).

Step2: Use the properties of a \(30 - 60-90\) triangle

In a \(30 - 60 - 90\) triangle, if the side adjacent to the \(60^{\circ}\) angle (\(GK\)) is \(x\), and the side opposite to the \(60^{\circ}\) angle (\(YK\)) is \(x\sqrt{3}\). Here \(x = 27\), so \(YK=27\sqrt{3}\).

Step3: Calculate \(\tan(G)\)

\(\tan(G)=\frac{YK}{GK}\), substituting \(YK = 27\sqrt{3}\) and \(GK = 27\), we get \(\tan(G)=\frac{27\sqrt{3}}{27}=\sqrt{3}\)

Answer:

\(\sqrt{3}\)