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.
A. 10.2 inches
B. 24.0 inches
C. 28.2 inches
D. 30.0 inches
Answer (1)
Calculate the length of the hypotenuse when the given sides are legs: 1 2 2 + 1 5 2 = 369 ≈ 19.2 .
Calculate the length of the leg when the hypotenuse and one leg are given: 1 5 2 − 1 2 2 = 81 = 9 .
Find the difference between the two possible lengths: ∣19.2 − 9∣ = 10.2 .
The difference between the two possible lengths of the third side is 10.2 inches .
Explanation
Problem Analysis We are given a right triangle with two sides of lengths 12 inches and 15 inches. We need to find the difference between the two possible lengths of the third side. Let's analyze the two possible cases.
Case 1: Hypotenuse Case 1: The third side is the hypotenuse. In this case, the two given sides are the legs of the right triangle. We can use the Pythagorean theorem to find the length of the hypotenuse, c , where a = 12 and b = 15 .
c = a 2 + b 2 = 1 2 2 + 1 5 2 = 144 + 225 = 369 c ≈ 19.2 inches
Case 2: Leg Case 2: The third side is a leg. In this case, the hypotenuse is 15 inches, and one leg is 12 inches. We can use the Pythagorean theorem to find the length of the other leg, a , where c = 15 and b = 12 .
a = c 2 − b 2 = 1 5 2 − 1 2 2 = 225 − 144 = 81 = 9 inches
Calculate the Difference Now, we need to find the difference between the two possible lengths of the third side. Difference = ∣19.2 − 9∣ = 10.2 inches
Final Answer Therefore, the difference between the two possible lengths of the third side of the triangle is approximately 10.2 inches.
Examples
Understanding right triangles and the Pythagorean theorem is crucial in many real-world applications. For example, when constructing a building, engineers use these principles to ensure that walls are perfectly vertical and corners are square. They might know the length of one side of a triangular structure and the hypotenuse and need to calculate the length of the other side to ensure structural integrity. This ensures the building is stable and safe.
