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?
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Detailed Explanation
The area of a triangle is \( \frac{\mathrm{1} }{\mathrm{2}}\)bh.
Let b represent the length of one leg.
Then h = 3b so the area is \( \frac{\mathrm{1} }{\mathrm{2}}\)bh = \( \frac{\mathrm{1} }{\mathrm{2}}\)2 × b × 3b = \( \frac{\mathrm{3} }{\mathrm{2}}\)\(b^2\) = 24, so \(2/3 × 3/2b^2 = 16\) and \(b^2 = 16\).
b = \( \sqrt{16}\) = 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 = \( \sqrt{160}\) = 12.6.
Let b represent the length of one leg.
Then h = 3b so the area is \( \frac{\mathrm{1} }{\mathrm{2}}\)bh = \( \frac{\mathrm{1} }{\mathrm{2}}\)2 × b × 3b = \( \frac{\mathrm{3} }{\mathrm{2}}\)\(b^2\) = 24, so \(2/3 × 3/2b^2 = 16\) and \(b^2 = 16\).
b = \( \sqrt{16}\) = 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 = \( \sqrt{160}\) = 12.6.
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