Question

find $q$.
write your answer in simplest radical form.
____ miles

Explanation:

Step1: Identify triangle type

This is a 45-45-90 right triangle, so the two legs are congruent, and the hypotenuse $c = a\sqrt{2}$ where $a$ is a leg length.

Step2: Set up hypotenuse formula

Let $q$ be the leg length. The hypotenuse is $3\sqrt{6}$, so:
$$3\sqrt{6} = q\sqrt{2}$$

Step3: Solve for $q$

Isolate $q$ by dividing both sides by $\sqrt{2}$, then rationalize:
$$q = \frac{3\sqrt{6}}{\sqrt{2}} = 3\sqrt{\frac{6}{2}} = 3\sqrt{3}$$

Answer:

$3\sqrt{3}$ miles