Question

the hypotenuse of an isosceles triangles measures 10 inches long. what is the length of one leg of the triangle?
\frac{10}{\sqrt{3}}
\frac{10}{\sqrt{2}}
10\sqrt{2}
10\sqrt{3}

Explanation:

Step1: Apply Pythagorean theorem

Let the length of each leg be \(x\). For an isosceles right - triangle (by definition of isosceles triangle with hypotenuse, it's a right - isosceles triangle), by Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), here \(a = b=x\) and \(c = 10\). So \(x^{2}+x^{2}=10^{2}\).

Step2: Simplify the equation

\(2x^{2}=100\), then \(x^{2} = 50\), \(x=\sqrt{50}\). Simplify \(\sqrt{50}=\sqrt{25\times2}=5\sqrt{2}\). Also, using the property of a \(45 - 45-90\) triangle (isosceles right - triangle), if the hypotenuse \(c\) and legs \(a = b\), then \(a=b=\frac{c}{\sqrt{2}}\). Substituting \(c = 10\), we get \(x=\frac{10}{\sqrt{2}}\).

Answer:

\(\frac{10}{\sqrt{2}}\)