Question

in \\( \triangle ghi, h = 820 \\) inches, \\( m\angle g = 102 ^ { \circ } \\) and \\( m\angle h = 10 ^ { \circ } \\). find the length of \\( g \\), to the nearest inch.

Explanation:

Step1: Find the measure of angle \(I\)

In a triangle, the sum of angles is \(180^{\circ}\). So, \(m\angle I=180^{\circ}-(m\angle G + m\angle H)\).
Substitute \(m\angle G = 102^{\circ}\) and \(m\angle H=10^{\circ}\):
\(m\angle I=180^{\circ}-(102^{\circ}+10^{\circ})=68^{\circ}\)

Step2: Use the Law of Sines

The Law of Sines states that \(\frac{g}{\sin G}=\frac{h}{\sin H}\).
We know \(h = 820\) inches, \(m\angle G = 102^{\circ}\), \(m\angle H=10^{\circ}\).
From \(\frac{g}{\sin G}=\frac{h}{\sin H}\), we can solve for \(g\): \(g=\frac{h\times\sin G}{\sin H}\)
Substitute the values: \(g=\frac{820\times\sin(102^{\circ})}{\sin(10^{\circ})}\)
We know that \(\sin(102^{\circ})\approx0.9781\) and \(\sin(10^{\circ})\approx0.1736\)
\(g=\frac{820\times0.9781}{0.1736}\)
First, calculate \(820\times0.9781 = 820\times(1 - 0.0219)=820-820\times0.0219=820 - 17.958=802.042\)
Then, \(g=\frac{802.042}{0.1736}\approx4619\)

Answer:

\(4619\)