Question

find the length of the third side. if necessary, write in simplest radical form.
9
2√14

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 hypotenuse \(c\) and legs \(a\) and \(b\), \(c^{2}=a^{2}+b^{2}\) (if \(c\) is the hypotenuse) or if we want to find a leg, \(a^{2}=c^{2}-b^{2}\) (where \(c\) is the hypotenuse and \(b\) is the other leg). Here, the hypotenuse \(c = 9\) and one leg \(b=2\sqrt{14}\), and we want to find the other leg \(a\).

Step2: Apply the Pythagorean theorem

Using the formula \(a=\sqrt{c^{2}-b^{2}}\), substitute \(c = 9\) and \(b = 2\sqrt{14}\) into the formula.
First, calculate \(c^{2}=9^{2}=81\) and \(b^{2}=(2\sqrt{14})^{2}=2^{2}\times(\sqrt{14})^{2}=4\times14 = 56\).
Then, \(a=\sqrt{81 - 56}=\sqrt{25}\).

Step3: Simplify the square root

Since \(\sqrt{25}=5\), the length of the third side is 5.

Answer:

5