Question

solving for side lengths of right triangles
determining missing components of right triangles
use the information given in the diagram of △ghi to find the missing measurements. round all answers to the nearest tenth.
m∠g = 51
hi =
gh = 18.9 23.3 47.7

Explanation:

Step1: Identify trigonometric ratio

In right - triangle $\triangle GHI$ with right - angle at $H$, we know $\angle I = 39^{\circ}$ and $GI = 30$. To find $GH$, we use the sine function since $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\sin I=\frac{GH}{GI}$.

Step2: Substitute values

We have $\sin(39^{\circ})=\frac{GH}{30}$, so $GH = 30\times\sin(39^{\circ})$.
Since $\sin(39^{\circ})\approx0.6293$, then $GH=30\times0.6293 = 18.879\approx18.9$.

Step3: To find $HI$

We use the cosine function since $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\cos I=\frac{HI}{GI}$.
Substitute $I = 39^{\circ}$ and $GI = 30$, so $HI = 30\times\cos(39^{\circ})$.
Since $\cos(39^{\circ})\approx0.7771$, then $HI=30\times0.7771 = 23.313\approx23.3$.

Answer:

$HI = 23.3$, $GH = 18.9$