Question

law of sines: (\frac{sin(a)}{a} = \frac{sin(b)}{b} = \frac{sin(c)}{c}) in (\triangle fgh), (h = 10), (mangle f = 65^circ), and (mangle g = 35^circ). what is the length of (g)? use the law of sines to find the answer. 9.2 units 9.8 units 6.7 units 5.8 units

Explanation:

Step1: Find angle at H

In a triangle, sum of angles is \(180^\circ\). So \(m\angle H = 180^\circ - 65^\circ - 35^\circ = 80^\circ\) (assuming the angle labeled \(P\) is a typo and should be \(F\), so \(m\angle F = 65^\circ\), \(m\angle G = 35^\circ\), side \(h\) is opposite \(\angle H\), side \(g\) is opposite \(\angle G\)).

Step2: Apply Law of Sines

Law of Sines: \(\frac{\sin(G)}{g}=\frac{\sin(H)}{h}\). We know \(h = 10\), \(m\angle G = 35^\circ\), \(m\angle H = 80^\circ\). So \(\frac{\sin(35^\circ)}{g}=\frac{\sin(80^\circ)}{10}\).

Step3: Solve for \(g\)

Cross - multiply: \(g=\frac{10\times\sin(35^\circ)}{\sin(80^\circ)}\). Calculate \(\sin(35^\circ)\approx0.5736\), \(\sin(80^\circ)\approx0.9848\). Then \(g=\frac{10\times0.5736}{0.9848}\approx\frac{5.736}{0.9848}\approx5.8\) units.

Answer:

5.8 units