Question

what is the perimeter, in meters, of the isosceles right triangle show below, whose hypotenuse is 10√2 meters long?

Explanation:

Step1: Find the length of the legs

In an isosceles right - triangle (a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle), the ratio of the sides is \(a:a:a\sqrt{2}\), where \(a\) is the length of each leg and \(a\sqrt{2}\) is the length of the hypotenuse.
Given that the hypotenuse \(c = 10\sqrt{2}\). If \(c=a\sqrt{2}\), then \(a\sqrt{2}=10\sqrt{2}\). Solving for \(a\), we divide both sides of the equation by \(\sqrt{2}\). So, \(a = 10\) meters.

Step2: Calculate the perimeter

The perimeter \(P\) of a triangle is \(P=a + b + c\). In an isosceles right - triangle \(a = b\).
We know \(a=b = 10\) and \(c=10\sqrt{2}\). Then \(P=10 + 10+10\sqrt{2}\).
Simplify the expression: \(P=(20 + 10\sqrt{2})\) meters.

Answer:

\((20 + 10\sqrt{2})\) meters