Question

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

Explanation:

Step1: Apply Pythagorean theorem

Let the third side be \(x\). According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the hypotenuse). Here \(c = 10\), \(a=4\), and \(b=x\). So \(4^{2}+x^{2}=10^{2}\).

Step2: Simplify the equation

\(16+x^{2}=100\). Then \(x^{2}=100 - 16\).

Step3: Calculate \(x^{2}\)

\(x^{2}=84\).

Step4: Find \(x\)

\(x=\sqrt{84}=\sqrt{4\times21}=2\sqrt{21}\)

Answer:

\(2\sqrt{21}\)