Question

solve the triangle.
write each answer as an integer or as a decimal rounded to the nearest tenth.

Explanation:

Step1: Find angle S

Sum of triangle angles is $180^\circ$.
$\angle S = 180^\circ - 92^\circ - 31^\circ = 57^\circ$

Step2: Calculate side $r$ (Law of Sines)

Relate $r$, $\angle R$, side 16, $\angle S$.
$\frac{r}{\sin(31^\circ)} = \frac{16}{\sin(57^\circ)}$
$r = \frac{16 \times \sin(31^\circ)}{\sin(57^\circ)} \approx \frac{16 \times 0.5150}{0.8387} \approx 9.8$

Step3: Calculate side $s$ (Law of Sines)

Relate $s$, $\angle Q$, side 16, $\angle S$.
$\frac{s}{\sin(92^\circ)} = \frac{16}{\sin(57^\circ)}$
$s = \frac{16 \times \sin(92^\circ)}{\sin(57^\circ)} \approx \frac{16 \times 0.9994}{0.8387} \approx 19.0$

Answer:

$\angle S = 57^\circ$, $r \approx 9.8$, $s \approx 19.0$