find the missing side

Explanation:

Step1: Apply Pythagorean theorem

In a right - triangle, if the hypotenuse is $c = 78$ and one side is $a = 30$, and the missing side is $b$. According to the Pythagorean theorem $c^{2}=a^{2}+b^{2}$, so $b=\sqrt{c^{2}-a^{2}}$.

Step2: Substitute values

Substitute $c = 78$ and $a = 30$ into the formula: $b=\sqrt{78^{2}-30^{2}}=\sqrt{(78 + 30)(78 - 30)}$ (using the difference - of - squares formula $x^{2}-y^{2}=(x + y)(x - y)$). Then $b=\sqrt{108\times48}=\sqrt{5184}=72$.

Answer:

72