Question Two sides of a right triangle have lengths of 5 inches and 12 inches. Which of the following could be the length, in inches, of the third side of the triangle?

In a right triangle, the lengths of the sides must follow the Pythagorean theorem, \(a^2 + b^2 = c^2\), where \(c\) is the length of the hypotenuse. Given sides 5 inches and 12 inches, we can find the hypotenuse: \(5^2 + 12^2 = 25 + 144 = 169\), so \(c = \sqrt{169} = 13\). Thus, one possibility for the third side is 13 inches. Additionally, the third side must also be longer than the difference of the two given sides, which is \(12 - 5 = 7\), and shorter than the sum, \(5 + 12 = 17\). So, possible lengths for the third side fall in the range of 7 inches to 13 inches (inclusive). To avoid any mix-ups, remember these tips: always check that the side lengths you choose don't violate the triangle inequality rules! For instance, if you mistakenly say the third side could be longer than 17 inches, you’d have a triangle that doesn’t exist! Keep your triangle sturdy by sticking to these principles!