Final answer:
By applying the Pythagorean theorem, it is concluded that the triangle with sides of lengths 24 miles, 32 miles, and 40 miles is a right triangle because the sum of the squares of the two shorter sides equals the square of the longest side (24² + 32² = 40²).
Step-by-step explanation:
To determine if a triangle with sides of lengths 24 miles, 32 miles, and 40 miles is a right triangle, we can apply the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the lengths of the two legs (a and b) is equal to the square of the length of the hypotenuse (c). This theorem is mathematically represented as a² + b² = c².
Let's apply this theorem:
Identify the potential hypotenuse, which, in this case, would be the longest side of the triangle: 40 miles.Calculate the square of the lengths of the other two sides: 24² = 576 and 32² = 1024.Add these sums: 576 + 1024 = 1600.Check if this sum equals the square of the potential hypotenuse: 40² = 1600.Since both are equal (1600 = 1600), the triangle is a right triangle according to the Pythagorean theorem.
The answer is A. Yes, the triangle with sides of lengths 24 miles, 32 miles, and 40 miles is a right triangle.