Question

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 = hi = gh = 39° 30

Explanation:

Step1: Find angle G

In a right - triangle, the sum of the two non - right angles is 90°. Given one non - right angle is 39°.
$m\angle G=90^{\circ}- 39^{\circ}=51^{\circ}$

Step2: Find side HI

We know $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For $\angle I = 39^{\circ}$ and adjacent side $GH$, $\tan39^{\circ}=\frac{HI}{GH}$. Since $GH$ is unknown for now, we can also use $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For $\angle I = 39^{\circ}$ and hypotenuse $GI = 30$. $\sin39^{\circ}=\frac{HI}{30}$, so $HI = 30\times\sin39^{\circ}\approx30\times0.6293\approx18.9$

Step3: Find side GH

We use $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. For $\angle I=39^{\circ}$ and hypotenuse $GI = 30$. $\cos39^{\circ}=\frac{GH}{30}$, so $GH = 30\times\cos39^{\circ}\approx30\times0.7771\approx23.3$

Answer:

$m\angle G = 51^{\circ}$
$HI\approx18.9$
$GH\approx23.3$