Question

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

Explanation:

Step1: Recall tangent definition

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

Step2: Analyze 30-60-90 triangle ratios

In a 30-60-90 triangle, sides are in ratio $1 : \sqrt{3} : 2$. $\angle Y = 30^\circ$, so $GK$ (opposite 30°) is shorter leg, $YK$ (opposite 60°) is longer leg. Longer leg = shorter leg $\times \sqrt{3}$, so $YK = 27\sqrt{3}$? Wait, no: Wait, $\angle G = 60^\circ$, so opposite to $G$ is $YK$, adjacent is $GK$. Wait, $\angle Y = 30^\circ$, so hypotenuse $GY = 2 \times GK = 54$ (since $GK$ is opposite 30°). Then $YK = \sqrt{GY^2 - GK^2} = \sqrt{54^2 - 27^2} = 27\sqrt{3}$. Then $\tan(G) = \frac{YK}{GK} = \frac{27\sqrt{3}}{27} = \sqrt{3}$. Wait, no, wait: $\angle G$ is 60°, so $\tan(60^\circ) = \sqrt{3}$. Alternatively, using 30-60-90 ratios: for angle 60°, opposite is $\sqrt{3}$ part, adjacent is 1 part. So $\tan(60^\circ) = \sqrt{3}$.

Answer:

$\sqrt{3}$ (the last option)