Question

in the right triangle, find the length of the side not given. give an exact answer and an approximation to three decimal places.
a = 6, b = 7

Explanation:

Step1: Apply Pythagorean theorem

In a right - triangle, \(c^{2}=a^{2}+b^{2}\) (where \(c\) is the hypotenuse and \(a,b\) are the legs). Given \(a = 6\) and \(b=7\), we substitute into the formula: \(c=\sqrt{a^{2}+b^{2}}\).

Step2: Calculate the exact value

\(c=\sqrt{6^{2}+7^{2}}=\sqrt{36 + 49}=\sqrt{85}\).

Step3: Calculate the approximate value

Using a calculator, \(\sqrt{85}\approx9.219544457\approx9.220\) (rounded to three decimal places).

Answer:

The exact length of the side \(c\) is \(\sqrt{85}\), and the approximate length is \(9.220\).