Question

examine the triangle. solve $x$ and name which tool you use.

Explanation:

Step1: Find third angle

First, calculate the missing angle in the triangle. The sum of angles in a triangle is $180^\circ$.
$\text{Third angle} = 180^\circ - 52^\circ - 63^\circ = 65^\circ$

Step2: Apply Law of Sines

Use the Law of Sines: $\frac{x}{\sin 63^\circ} = \frac{8}{\sin 65^\circ}$
Rearrange to solve for $x$:
$x = \frac{8 \times \sin 63^\circ}{\sin 65^\circ}$

Step3: Calculate the value

Use approximate sine values: $\sin 63^\circ \approx 0.8910$, $\sin 65^\circ \approx 0.9063$
$x \approx \frac{8 \times 0.8910}{0.9063} \approx \frac{7.128}{0.9063} \approx 7.86$

Answer:

$x \approx 7.9$ miles (rounded to one decimal place). The tool used is the Law of Sines.