Question

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

Explanation:

Step1: Identify the right - angled triangle formula

We use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (the longest side) and \(a\) and \(b\) are the other two sides. In a right - angled triangle, if the hypotenuse \(c = 25\) and one side \(a=7\), and we want to find the other side \(b\). We can rewrite the formula as \(b=\sqrt{c^{2}-a^{2}}\).

Step2: Substitute the values

Substitute \(c = 25\) and \(a = 7\) into the formula \(b=\sqrt{25^{2}-7^{2}}=\sqrt{(25 + 7)(25 - 7)}\) (using the difference - of - squares formula \(x^{2}-y^{2}=(x + y)(x - y)\)). First, calculate \(25^{2}=625\) and \(7^{2}=49\). Then \(b=\sqrt{625-49}=\sqrt{576}\).

Step3: Calculate the square root

\(\sqrt{576}=24\).

Answer:

24