Question

the lengths of two sides of a right triangle are 12 inches and 15 inches. what is the difference between the two possible lengths of the third side of the triangle? round your answer to the nearest tenth. 10.2 inches 24.0 inches 28.2 inches 30.0 inches

Explanation:

Step1: Determine the two - case scenarios

In a right - triangle, if \(a\), \(b\) are the legs and \(c\) is the hypotenuse, then \(c^{2}=a^{2}+b^{2}\) (when \(c\) is the hypotenuse), and if one of the given sides is the hypotenuse, say \(c\) and the other two are \(a\) and \(b\), then \(b^{2}=c^{2}-a^{2}\).

Step2: Case 1: Hypotenuse is the unknown side

Let \(a = 12\) and \(b = 15\). Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}=\sqrt{12^{2}+15^{2}}=\sqrt{144 + 225}=\sqrt{369}\approx19.2\).

Step3: Case 2: One of the given sides is the hypotenuse

Let the hypotenuse \(c = 15\) and one leg \(a = 12\). Then the other leg \(b=\sqrt{c^{2}-a^{2}}=\sqrt{15^{2}-12^{2}}=\sqrt{225 - 144}=\sqrt{81}=9\).

Step4: Calculate the difference

The difference between the two possible lengths of the third side is \(19.2−9 = 10.2\).

Answer:

10.2 inches