Question

the lengths of two sides of a right triangle are 5 inches and 8 inches. what is the difference between the two possib lengths of the third side of the triangle? round your answer to the nearest tenth.
2.4 inches
3.2 inches
10.0 inches
15.7 inches

Explanation:

Step1: Calculate the third side when it is a leg

Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse. If the hypotenuse \(c = 8\) and one leg \(a=5\), then the other leg \(b=\sqrt{c^{2}-a^{2}}\).

$$b=\sqrt{8^{2}-5^{2}}=\sqrt{64 - 25}=\sqrt{39}\approx6.2$$

Step2: Calculate the third side when it is the hypotenuse

If the two legs are \(a = 5\) and \(b=8\), then the hypotenuse \(c=\sqrt{a^{2}+b^{2}}\)

$$c=\sqrt{5^{2}+8^{2}}=\sqrt{25 + 64}=\sqrt{89}\approx9.4$$

Step3: Find the difference between the two possible lengths

The difference is \(9.4-6.2 = 3.2\)

Answer:

3.2 inches