Question

question find the length of the third side. if necessary, write in simplest radical form. answer attempt 1 out of 2

Explanation:

Step1: Apply the Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse. Let \(a = 5\), \(c=5\sqrt{5}\), and we need to find \(b\). Then \(b^{2}=c^{2}-a^{2}\).
Substitute the values: \(b^{2}=(5\sqrt{5})^{2}-5^{2}\).

Step2: Simplify the expression

First, calculate \((5\sqrt{5})^{2}=5^{2}\times(\sqrt{5})^{2}=25\times5 = 125\) and \(5^{2}=25\).
So \(b^{2}=125 - 25\).
\(b^{2}=100\).

Step3: Solve for \(b\)

Take the square root of both sides: \(b=\sqrt{100}\).
Since \(b>0\) (length of a side of a triangle), \(b = 10\).

Answer:

\(10\)