Question

write the tangent ratios for $\angle y$ and $\angle z$:

not drawn to scale

$\tan y=\frac{9}{4},\tan z=\frac{4}{9}$

$\tan y=\frac{4}{\sqrt{97}},\tan z=\frac{9}{\sqrt{97}}$

$\tan y=\frac{\sqrt{97}}{4},\tan z=\frac{\sqrt{97}}{9}$

$\tan y=\frac{4}{9},\tan z=\frac{9}{4}$

Explanation:

Step1: Recall the tangent ratio formula

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Find \(\tan Y\)

For \(\angle Y\), the opposite side is \(XZ = 4\) and the adjacent side is \(XY=9\). So, \(\tan Y=\frac{XZ}{XY}=\frac{4}{9}\).

Step3: Find \(\tan Z\)

For \(\angle Z\), the opposite side is \(XY = 9\) and the adjacent side is \(XZ = 4\). So, \(\tan Z=\frac{XY}{XZ}=\frac{9}{4}\).

Answer:

\(\tan Y=\frac{4}{9},\tan Z=\frac{9}{4}\) (the last option)