Final answer:
To determine if a set of side lengths can form a right triangle, we can check if the sum of the squares of the two shorter sides is equal to the square of the longest side. Out of the given options, the side lengths that cannot form a right triangle are: 24, 23, and 40.
Step-by-step explanation:
A right triangle is a triangle that has one angle measuring 90 degrees. In a right triangle, the side lengths must satisfy the Pythagorean theorem, which states that 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.
To determine if a set of side lengths can form a right triangle, we can check if the sum of the squares of the two shorter sides is equal to the square of the longest side. Let's go through each set of side lengths:
- a. Side lengths: 24, 23, and 40
The sum of squares of the shorter sides: 24^2 + 23^2 = 1156 + 529 = 1685
The square of the longest side: 40^2 = 1600
1685 is not equal to 1600, so this set of side lengths cannot form a right triangle. - b. Side lengths: 37, 95, and 13
The sum of squares of the shorter sides: 37^2 + 13^2 = 1369 + 169 = 1538
The square of the longest side: 95^2 = 9025
1538 is not equal to 9025, so this set of side lengths cannot form a right triangle. - c. Side lengths: 85, 13, and 84
The sum of squares of the shorter sides: 85^2 + 13^2 = 7225 + 169 = 7394
The square of the longest side: 84^2 = 7056
7394 is not equal to 7056, so this set of side lengths cannot form a right triangle. - d. Side lengths: 7.4, 19, and 125
The sum of squares of the shorter sides: 7.4^2 + 19^2 = 54.76 + 361 = 415.76
The square of the longest side: 125^2 = 15625
415.76 is not equal to 15625, so this set of side lengths cannot form a right triangle.
Out of the given options, the side lengths that cannot form a right triangle are: a. 24, 23, and 40. Therefore, option a is the correct answer.