Question

find the length of the third side. if necessary round to the nearest tenth.

Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem. The Pythagorean theorem states that for a right - triangle with legs of length \(a\) and \(b\) and hypotenuse of length \(c\), \(c^{2}=a^{2}+b^{2}\) (if \(c\) is the hypotenuse) or \(b^{2}=c^{2}-a^{2}\) (if \(b\) is a leg and \(c\) is the hypotenuse). Here, we know one leg \(a = 7\) and the hypotenuse \(c = 25\), and we want to find the other leg \(b\). So we use the formula \(b=\sqrt{c^{2}-a^{2}}\).

Step2: Substitute the values into the formula

Substitute \(a = 7\) and \(c = 25\) into the formula \(b=\sqrt{c^{2}-a^{2}}\). First, calculate \(c^{2}-a^{2}\):
\(c^{2}-a^{2}=25^{2}-7^{2}\)
\(25^{2}=625\) and \(7^{2}=49\), so \(25^{2}-7^{2}=625 - 49=576\)

Step3: Take the square root

Then \(b=\sqrt{576}\)
\(\sqrt{576}=24\)

Answer:

The length of the third side is \(24\).