Question

find the length of the third side. if necessary, write in simplest radical form

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse. Let the unknown side be \(x\). Here, \(c = 2\sqrt{5}\), \(b = 4\). Then \(x^{2}+4^{2}=(2\sqrt{5})^{2}\).

Step2: Simplify the equation

\(x^{2}+16 = 4\times5\). So \(x^{2}+16 = 20\).

Step3: Solve for \(x\)

Subtract 16 from both sides: \(x^{2}=20 - 16\). Then \(x^{2}=4\). Take the square root of both sides, \(x=\sqrt{4}\). Since length is non - negative, \(x = 2\).

Answer:

\(2\)