Question

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

Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem, which states that for a right - triangle with legs \(a\), \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). Let the legs be \(a = 9\) and \(x\) (the side we want to find), and the hypotenuse \(c=\sqrt{97}\).

Step2: Apply the Pythagorean theorem

We know that \(a^{2}+x^{2}=c^{2}\), so \(x^{2}=c^{2}-a^{2}\). Substitute \(a = 9\) and \(c=\sqrt{97}\) into the formula:
\(x^{2}=(\sqrt{97})^{2}-9^{2}\)
\(x^{2}=97 - 81\)
\(x^{2}=16\)

Step3: Solve for \(x\)

Take the square root of both sides: \(x=\sqrt{16}=4\) (we take the positive root because length cannot be negative).

Answer:

\(4\)