the lengths of two sides of a right triangle are 5 inches and 8 inches. what is the difference between the…

Explanation:

Step1: Identify cases for third side

In a right triangle, the third side can be a leg or the hypotenuse. Let's denote the sides as (a = 5), (b = 8), and (c) (third side).

Case 1: 8 is the hypotenuse

Use Pythagorean theorem (a^2 + c^2 = b^2) (here (b = 8), (a = 5)). So (5^2 + c^2 = 8^2).
(25 + c^2 = 64)
(c^2 = 64 - 25 = 39)
(c = \sqrt{39} \approx 6.2) (rounded to nearest tenth)

Case 2: 8 is a leg (so hypotenuse is (c))

Use Pythagorean theorem (a^2 + b^2 = c^2) (here (a = 5), (b = 8)).
(5^2 + 8^2 = c^2)
(25 + 64 = c^2)
(c^2 = 89)
(c = \sqrt{89} \approx 9.4) (rounded to nearest tenth)

Step2: Find the difference

Subtract the two possible lengths: (9.4 - 6.2 = 3.2)

Answer:

3.2 inches (corresponding to the option "3.2 inches")