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: Find the third side when 15 is the hypotenuse

By Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), let \(c = 15\), \(a=12\), then \(b=\sqrt{c^{2}-a^{2}}\)
\(b=\sqrt{15^{2}-12^{2}}=\sqrt{225 - 144}=\sqrt{81}=9\)

Step2: Find the third side when it is the hypotenuse

Let the two legs be \(a = 12\) and \(b = 15\), then \(c=\sqrt{a^{2}+b^{2}}\)
\(c=\sqrt{12^{2}+15^{2}}=\sqrt{144 + 225}=\sqrt{369}\approx19.2\)

Step3: Calculate the difference

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

Answer:

10.2 inches