Which units of length represent the sides of a right triangle? Choose two answers. A. 3, 4, and 5 B. 6, 8, and 15 C. 20, 21, and 28 D. 9, 12, and 15.

The units of length that represent the sides of a right triangle can be determined using the Pythagorean theorem, which is a fundamental relation in Euclidean geometry among the three sides of a right triangle. The theorem states that:

c² = a² + b²

where c is the length of the hypotenuse (the side opposite the right angle), and a and b are the lengths of the other two sides.

Now, let's apply this theorem to the options provided:

A. 3, 4, and 5 We calculate the square of the largest number: 5² = 25, and the sum of the squares of the other two numbers: 3² + 4² = 9 + 16 = 25. Since 25 = 25, this set of numbers satisfies the Pythagorean theorem, which means they could represent the sides of a right triangle.

B. 6, 8, and 15 We calculate: 15² = 225, and 6² + 8² = 36 + 64 = 100. Since 225 ≠ 100, this set of numbers does not satisfy the Pythagorean theorem, meaning they cannot represent the sides of a right triangle.

C. 20, 21, and 28 We calculate: 28² = 784, and 20² + 21² = 400 + 441 = 841. Since 784 ≠ 841, this set of numbers does not satisfy the Pythagorean theorem, which means they cannot represent the sides of a right triangle.

D. 9, 12, and 15 We calculate: 15² = 225, and 9² + 12² = 81 + 144 = 225. Since 225 = 225, this set of numbers satisfies the Pythagorean theorem, which means they could represent the sides of a right triangle.

Therefore, the sets of numbers that can represent the sides of a right triangle are: A. 3, 4, and 5, and D. 9, 12, and 15.