Question

use the information given in the diagram of △ghi to find the missing measurements. round all answers to the nearest tenth. m∠g =
hi =
gh =

Explanation:

Step1: Find $HI$ using sine function

In right - triangle $GHI$, $\sin G=\frac{HI}{GI}$. Given $GI = 30$ and $\angle G=39^{\circ}$, then $HI = GI\times\sin G$. So $HI=30\times\sin39^{\circ}$. Using a calculator, $\sin39^{\circ}\approx0.6293$, and $HI = 30\times0.6293=18.9$.

Step2: Find $GH$ using cosine function

In right - triangle $GHI$, $\cos G=\frac{GH}{GI}$. Given $GI = 30$ and $\angle G = 39^{\circ}$, then $GH=GI\times\cos G$. So $GH = 30\times\cos39^{\circ}$. Using a calculator, $\cos39^{\circ}\approx0.7771$, and $GH=30\times0.7771 = 23.3$.

Step3: Angle sum property of a triangle

In $\triangle GHI$, $\angle H = 90^{\circ}$, $\angle G=39^{\circ}$, and we know that the sum of angles in a triangle is $180^{\circ}$. So $\angle I=180^{\circ}-\angle H-\angle G=180^{\circ}-90^{\circ}-39^{\circ}=51^{\circ}$.

Answer:

$HI = 18.9$, $GH = 23.3$, $m\angle I=51.0^{\circ}$