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.
Answer (1)
Calculate the length of the third side when it is the hypotenuse: 5 2 + 8 2 = 89 ≈ 9.43 .
Calculate the length of the third side when 8 is the hypotenuse: 8 2 − 5 2 = 39 ≈ 6.24 .
Find the difference between the two possible lengths: ∣9.43 − 6.24∣ = 3.19 .
Round the difference to the nearest tenth: 3.2 .
Explanation
Problem Analysis We are given a right triangle with two sides of lengths 5 inches and 8 inches. We need to find the difference between the two possible lengths of the third side. There are two cases to consider: either the third side is the hypotenuse, or the side with length 8 is the hypotenuse.
Case 1: Hypotenuse is the third side Case 1: The third side is the hypotenuse. Let a = 5 and b = 8 . Then, by the Pythagorean theorem, the length of the hypotenuse c is given by: c = a 2 + b 2 = 5 2 + 8 2 = 25 + 64 = 89 ≈ 9.43 So, the length of the third side is approximately 9.43 inches.
Case 2: 8 is the hypotenuse Case 2: The side with length 8 is the hypotenuse. Let c = 8 and a = 5 . Then, by the Pythagorean theorem, the length of the other side b is given by: b = c 2 − a 2 = 8 2 − 5 2 = 64 − 25 = 39 ≈ 6.24 So, the length of the third side is approximately 6.24 inches.
Calculate the difference Now, we find the difference between the two possible lengths of the third side: Difference = ∣ 89 − 39 ∣ ≈ ∣9.43 − 6.24∣ = 3.19 Rounding to the nearest tenth, the difference is 3.2 inches.
Final Answer The difference between the two possible lengths of the third side is approximately 3.2 inches.
Examples
Understanding right triangles is crucial in construction. For example, when building a ramp, you need to ensure it forms a right triangle with the ground and the vertical support. If you know the lengths of two sides (like the base and the height), you can calculate the length of the ramp (the hypotenuse) using the Pythagorean theorem. This ensures the ramp is stable and meets the required slope for accessibility.
