Question

find the missing side length. keep answers as a simplified radical if not whole numbers. 15 mi x 16 mi x = miles do the side lengths form a pythagorean triple? (yes or no)

Explanation:

Step1: Apply Pythagorean theorem

In a right - triangle, \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (the longest side). Here, assume the hypotenuse is \(16\) mi, and the other two sides are \(x\) and \(15\) mi. So \(x^{2}+15^{2}=16^{2}\).

Step2: Solve for \(x^{2}\)

\(x^{2}=16^{2}-15^{2}\). Using the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\), we have \(x^{2}=(16 + 15)(16-15)=(31)\times(1)=31\).

Step3: Solve for \(x\)

\(x=\sqrt{31}\) mi.

Step4: Check Pythagorean triple

A Pythagorean triple consists of three positive integers \(a\), \(b\), and \(c\) such that \(a^{2}+b^{2}=c^{2}\). Since \(\sqrt{31}\) is not an integer, the side - lengths do not form a Pythagorean triple.

Answer:

\(x = \sqrt{31}\) miles
No