Question

in \\( delta ghi, g = 75 \\) inches, \\( m angle h = 151 ^ { circ } \\) and \\( m angle i = 7 ^ { circ } \\). find the length of \\( h \\), to the nearest 10th of an inch.

Explanation:

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

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

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{h}{\sin H}=\frac{g}{\sin G}\).
We know \(g = 75\) inches, \(m\angle H=151^{\circ}\), \(m\angle G = 22^{\circ}\).
From \(\frac{h}{\sin H}=\frac{g}{\sin G}\), we can solve for \(h\): \(h=\frac{g\times\sin H}{\sin G}\).
Substitute the values: \(h=\frac{75\times\sin(151^{\circ})}{\sin(22^{\circ})}\).
We know that \(\sin(151^{\circ})=\sin(180 - 29)^{\circ}=\sin(29^{\circ})\approx0.4848\) and \(\sin(22^{\circ})\approx0.3746\).
Then \(h=\frac{75\times0.4848}{0.3746}=\frac{36.36}{0.3746}\approx97.1\).

Answer:

\(h\approx97.1\) inches.