Question

in $\triangle fgh$, $f = 8.8$ inches, $mangle f = 23^{circ}$ and $mangle g = 107^{circ}$. find the length of $g$, to the nearest 10th of an inch.

Explanation:

Step1: Find angle H

The sum of angles in a triangle is 180°. So, $m\angle H=180^{\circ}-m\angle F - m\angle G=180^{\circ}-23^{\circ}-107^{\circ}=50^{\circ}$.

Step2: Use the Law of Sines

The Law of Sines states that $\frac{f}{\sin F}=\frac{g}{\sin G}$. We know $f = 8.8$, $\sin F=\sin(23^{\circ})$, and $\sin G=\sin(107^{\circ})$. Rearranging for $g$ gives $g=\frac{f\times\sin G}{\sin F}$.

Step3: Substitute values

$g=\frac{8.8\times\sin(107^{\circ})}{\sin(23^{\circ})}$. We know that $\sin(107^{\circ})\approx0.9563$ and $\sin(23^{\circ})\approx0.3907$. Then $g=\frac{8.8\times0.9563}{0.3907}=\frac{8.41544}{0.3907}\approx21.5$.

Answer:

$21.5$