Question

in $\triangle ghi$, $g = 75$ inches, $mangle h = 151^{circ}$ and $mangle i = 7^{circ}$. find the length of $h$, to the nearest 10th of an inch.

Explanation:

Step1: Find angle G

The sum of angles in a triangle is 180°. So, $m\angle G=180^{\circ}-(m\angle H + m\angle I)=180^{\circ}-(151^{\circ}+7^{\circ}) = 22^{\circ}$.

Step2: Use the Law of Sines

The Law of Sines states that $\frac{h}{\sin H}=\frac{g}{\sin G}$. We know $g = 75$ inches, $m\angle H=151^{\circ}$, and $m\angle G = 22^{\circ}$. Rearranging for $h$ gives $h=\frac{g\sin H}{\sin G}$.

Step3: Calculate h

Substitute the values: $h=\frac{75\times\sin(151^{\circ})}{\sin(22^{\circ})}$. Since $\sin(151^{\circ})\approx0.4848$ and $\sin(22^{\circ})\approx0.3746$, then $h=\frac{75\times0.4848}{0.3746}=\frac{36.36}{0.3746}\approx97.1$.

Answer:

$97.1$ inches