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. 5.8 units 6.7 units 9.2 units 9.8 units

Explanation:

Step1: Find angle $H$

The sum of angles in a triangle is $180^{\circ}$. So $m\angle H=180^{\circ}-(m\angle F + m\angle G)=180^{\circ}-(65^{\circ}+ 35^{\circ}) = 80^{\circ}$.

Step2: Apply the law of sines

The law of sines states $\frac{\sin(F)}{f}=\frac{\sin(G)}{g}=\frac{\sin(H)}{h}$. We want to find $g$, and we know $h = 10$, $m\angle G=35^{\circ}$, $m\angle H = 80^{\circ}$. So $\frac{\sin(G)}{g}=\frac{\sin(H)}{h}$, which can be rewritten as $g=\frac{h\sin(G)}{\sin(H)}$.

Step3: Substitute values and calculate

$g=\frac{10\times\sin(35^{\circ})}{\sin(80^{\circ})}$. Since $\sin(35^{\circ})\approx0.5736$ and $\sin(80^{\circ})\approx0.9848$, then $g=\frac{10\times0.5736}{0.9848}\approx 5.8$.

Answer:

5.8 units