Which of the following sets of three lengths could be the sides of a right triangle?
19 cm, 98 cm, 100 cm
20 cm, 99 cm, 101 cm
21 cm, 100 cm, 102 cm
22 cm, 101 cm, 103 cm
To determine whether a set of three lengths can form the sides of a right triangle, you can use the Pythagorean theorem. This theorem 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 (the legs). In other words, for lengths `a`, `b`, and `c`, where `c` is the hypothenuse,
`c^2 = a^2 + b^2`.
Let's apply this test to each set of lengths:
1. For sides 19 cm, 98 cm, 100 cm:
`100^2 = 19^2 + 98^2`
`10000 = 361 + 9604`
`10000 = 9965` (which does not satisfy the Pythagorean theorem)
2. For sides 20 cm, 99 cm, 101 cm:
`101^2 = 20^2 + 99^2`
`10201 = 400 + 9801`
`10201 = 10201` (which satisfies the Pythagorean theorem)
3. For sides 21 cm, 100 cm, 102 cm:
`102^2 = 21^2 + 100^2`
`10404 = 441 + 10000`
`10404 = 10441` (which does not satisfy the Pythagorean theorem)
4. For sides 22 cm, 101 cm, 103 cm:
`103^2 = 22^2 + 101^2`
`10609 = 484 + 10201`
`10609 = 10685` (which does not satisfy the Pythagorean theorem)
Based on the Pythagorean theorem, the only set of lengths that could be the sides of a right triangle is 20 cm, 99 cm, and 101 cm.