Question

a sailboat is 12 km north of a lighthouse. a motor cruiser is 12 km east of the same lighthouse. use trigonometry to find an exact expression for the distance between the two boats. your answer is:

Explanation:

Step1: Identify the right - angled triangle

The positions of the sailboat, motor - cruiser, and lighthouse form a right - angled triangle. The distance of the sailboat from the lighthouse is one leg ($a = 12$ km) and the distance of the motor - cruiser from the lighthouse is the other leg ($b = 12$ km). The distance between the two boats is the hypotenuse $c$.

Step2: Apply the Pythagorean theorem

According to the Pythagorean theorem $c^{2}=a^{2}+b^{2}$. Substituting $a = 12$ and $b = 12$ into the formula, we get $c^{2}=12^{2}+12^{2}=144 + 144=288$.

Step3: Solve for $c$

Taking the square root of both sides, $c=\sqrt{288}=\sqrt{144\times2}=12\sqrt{2}$ km.

Answer:

$12\sqrt{2}$ km