Final answer:
After applying the Pythagorean theorem to the sides 13 inches, 57 inches, and 59 inches, we find that the sum of the squares of the shorter sides does not equal the square of the longest side (3418 ≠ 3481), indicating that the triangle is not a right triangle.
Step-by-step explanation:
To determine whether a triangle with sides of 13 inches, 57 inches, and 59 inches is a right triangle, we apply the Pythagorean theorem, which states that in a right triangle the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Here, c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
The longest side of the given triangle would be the hypotenuse if the triangle is a right triangle. So let's check this using the sides 13 inches, 57 inches, and 59 inches, with 59 inches being the hypothesized hypotenuse.
To test the Pythagorean relationship, we calculate:
a² + b² and compare it to c².
- a² = 13² = 169
- b² = 57² = 3249
- a² + b² = 169 + 3249 = 3418
- c² = 59² = 3481
Since a² + b² does not equal c² (3418 ≠ 3481), this set of side lengths does not satisfy the Pythagorean theorem. Therefore, the triangle with sides of 13 inches, 57 inches, and 59 inches is not a right triangle.