Question

the lengths of two sides of a right triangle are 5 inches and 8 inches. what is the difference between the two possible 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: Identify cases (legs/hypotenuse)

Case 1: 5 and 8 are legs. Use Pythagoras: \( c = \sqrt{5^2 + 8^2} = \sqrt{25 + 64} = \sqrt{89} \approx 9.43 \)
Case 2: 8 is hypotenuse, 5 is leg. Find other leg: \( b = \sqrt{8^2 - 5^2} = \sqrt{64 - 25} = \sqrt{39} \approx 6.24 \)

Step2: Calculate differences from options

Option 2.4: \( |9.43 - 6.24| \approx 3.19 \approx 3.2 \); \( |9.43 - 2.4| = 7.03 \); \( |6.24 - 2.4| = 3.84 \)
Option 3.2: Matches the first difference (≈3.2)
Option 10.0: \( |10 - 9.43| = 0.57 \); \( |10 - 6.24| = 3.76 \)
Option 15.7: Too large, differences >7

Answer:

3.2 inches (the option with 3.2 inches)