Question

the perimeter of a right isosceles triangle is 76 + 76√2. what is the length of the hypotenuse?

Explanation:

Step1: Let leg - length be \(x\).

In a right - isosceles triangle, the two legs have the same length \(x\), and by the Pythagorean theorem, the hypotenuse \(c=\sqrt{x^{2}+x^{2}}=\sqrt{2x^{2}}=\sqrt{2}x\). The perimeter \(P = 2x + c=2x+\sqrt{2}x=(2 + \sqrt{2})x\).

Step2: Set up the equation for the perimeter.

We know that \(P=76 + 76\sqrt{2}\), and \(P=(2+\sqrt{2})x\). So, \((2+\sqrt{2})x=76 + 76\sqrt{2}\).

Step3: Solve for \(x\).

Factor out 76 on the right - hand side: \((2+\sqrt{2})x=76(1 + \sqrt{2})\). Then \(x = 76\).

Step4: Find the length of the hypotenuse.

Since the hypotenuse \(c=\sqrt{2}x\), substituting \(x = 76\), we get \(c = 76\sqrt{2}\).

Answer:

\(76\sqrt{2}\)