Question

solve the triangle.
write each answer as an integer or as a decimal rounded to the nearest tenth.
m∠r =
r =
t =

Explanation:

Step1: Find angle $R$

The sum of angles in a triangle is $180^{\circ}$. So, $m\angle R=180^{\circ}-42^{\circ}-30^{\circ}=108^{\circ}$.

Step2: Use the Law of Sines to find $r$

The Law of Sines states that $\frac{\sin T}{t}=\frac{\sin S}{s}=\frac{\sin R}{r}$. We know $T = 42^{\circ}$, $S=30^{\circ}$, $R = 108^{\circ}$ and $s = 9$. So, $\frac{\sin R}{r}=\frac{\sin S}{s}$, substituting values: $\frac{\sin108^{\circ}}{r}=\frac{\sin30^{\circ}}{9}$. Then $r=\frac{9\sin108^{\circ}}{\sin30^{\circ}}=\frac{9\times0.9511}{0.5}\approx17.1$.

Step3: Use the Law of Sines to find $t$

Using $\frac{\sin T}{t}=\frac{\sin S}{s}$, substituting $T = 42^{\circ}$, $S = 30^{\circ}$ and $s=9$. So, $\frac{\sin42^{\circ}}{t}=\frac{\sin30^{\circ}}{9}$, then $t=\frac{9\sin42^{\circ}}{\sin30^{\circ}}=\frac{9\times0.6691}{0.5}\approx12.0$.

Answer:

$m\angle R = 108^{\circ}$
$r\approx17.1$
$t\approx12.0$