Question

1. two sides of a triangle measure 18 inches and 24 inches. what is the length of the third side if the side lengths are a pythagorean triple? (numerical value only)

Explanation:

Step1: Recall Pythagorean theorem

In a right - triangle, \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the hypotenuse, the longest side). We need to check if 18 and 24 can be part of a Pythagorean triple. First, factor out the greatest common divisor of 18 and 24, which is 6. So 18 = 6×3 and 24 = 6×4. A well - known Pythagorean triple is 3, 4, 5.

Step2: Calculate the third side

If the two given sides are the legs of the right - triangle, then the hypotenuse \(c=\sqrt{18^{2}+24^{2}}=\sqrt{324 + 576}=\sqrt{900}=30\). If 24 is the hypotenuse and 18 is one of the legs, then the other leg \(b=\sqrt{24^{2}-18^{2}}=\sqrt{(24 + 18)(24 - 18)}=\sqrt{42\times6}=\sqrt{252}\), and \(\sqrt{252}\) is not an integer. Since we want a Pythagorean triple (all integer side lengths), the third side is 30.

Answer:

30