Which unit lengths represent the sides of a right triangle? Select two answers.
A. 11, 60, and 61
B. 12, 35, and 37
C. 8, 13, and 13
D. 5, 7, and 9
A. 11, 60, and 61
B. 12, 35, and 37
The unit lengths that represent the sides of a right triangle are the ones that 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 other two sides.
Looking at the given options:
A. 11, 60, and 61
B. 12, 35, and 37
C. 8, 13, and 13
D. 5, 7, and 9
Option A (11, 60, and 61) satisfies the Pythagorean theorem since 11^2 + 60^2 = 3721 + 3600 = 7321, which is equal to 61^2.
Option B (12, 35, and 37) also satisfies the Pythagorean theorem since 12^2 + 35^2 = 144 + 1225 = 1369, which is equal to 37^2.
Option C (8, 13, and 13) does not satisfy the Pythagorean theorem since 8^2 + 13^2 = 64 + 169 = 233, which is not equal to 13^2.
Option D (5, 7, and 9) also does not satisfy the Pythagorean theorem since 5^2 + 7^2 = 25 + 49 = 74, which is not equal to 9^2.
Therefore, the correct answers are A. 11, 60, and 61 and B. 12, 35, and 37.
To determine which unit lengths represent the sides of a right triangle, we need to use the Pythagorean theorem. According to the theorem, in a right triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side, which is called the hypotenuse.
Let's calculate the squares of the lengths of the sides for each option:
A. For option A, the squares of the lengths are:
11^2 = 121
60^2 = 3600
61^2 = 3721
B. For option B, the squares of the lengths are:
12^2 = 144
35^2 = 1225
37^2 = 1369
C. For option C, the squares of the lengths are:
8^2 = 64
13^2 = 169
13^2 = 169
D. For option D, the squares of the lengths are:
5^2 = 25
7^2 = 49
9^2 = 81
According to the Pythagorean theorem, a right triangle should have one side length that is the square root of the sum of the squares of the other two side lengths.
Based on our calculations:
A. 11, 60, and 61 do not satisfy the theorem because 11^2 + 60^2 ≠ 61^2.
B. 12, 35, and 37 satisfy the theorem because 12^2 + 35^2 = 37^2.
C. 8, 13, and 13 do not satisfy the theorem because 8^2 + 13^2 ≠ 13^2.
D. 5, 7, and 9 satisfy the theorem because 5^2 + 7^2 = 9^2.
Therefore, the correct answer is B. 12, 35, and 37.