A right triangle has an area of 24 square feet. If one leg is 3 times as long as the other, what is the length of the longest side?

A right triangle has an area of 24 square feet. If one leg is 3 times as long as the other, what is the length of the longest side?

Detailed Explanation

The area of a triangle is 1/2 bh. Let b represent the length of one leg. Then h = 3b so the area is \(1/2bh = 1/2 × b × 3b = 3/2b^2 = 24\), so \(2/3 × 3/2b^2 = 16\) and \(b^2 = 16\). b = √16 = 4 and h = 3 × 4 = 12. The longest side of a right triangle is the hypotenuse. Using the Pythagorean theorem, \(leg^2 + leg^2 = hypotenuse^2\), so \(4^2 + 12^2 = c^2\) and \(16 + 144 = c^2\). Therefore, \(160 = c^2\) and c = √160 = 12.6.