Question

find the length of the missing side.

Explanation:

Step1: Identify the formula

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse and \(a\), \(b\) are the legs of the right - triangle. Here, \(c = 82\) in and one leg \(b = 18\) in. Let the missing side be \(a\). Then \(a=\sqrt{c^{2}-b^{2}}\).

Step2: Substitute the values

Substitute \(c = 82\) and \(b = 18\) into the formula: \(a=\sqrt{82^{2}-18^{2}}=\sqrt{(82 + 18)(82 - 18)}\) (using the difference - of - squares formula \(x^{2}-y^{2}=(x + y)(x - y)\)).

Step3: Calculate

First, \(82+18 = 100\) and \(82 - 18=64\). Then \(a=\sqrt{100\times64}=\sqrt{100}\times\sqrt{64}=10\times8 = 80\) in.

Answer:

80 in.