Final answer:
The sets of side lengths that represent a right triangle are a) 1, 1, √2, b) √2, √3, √5, and d) 7, 24, 25.
Step-by-step explanation:
To determine which of the following sets of side lengths represents the side lengths of a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two shorter sides (legs) is equal to the square of the length of the longest side (hypotenuse).
a) 1, 1, √2: Applying the Pythagorean theorem, we have 1² + 1² = (√2)², which simplifies to 2 = 2. Thus, this set of side lengths represents a right triangle.
b) √2, √3, √5: Applying the Pythagorean theorem, we have (√2)² + (√3)² = (√5)², which simplifies to 2 + 3 = 5. Thus, this set of side lengths represents a right triangle.
c) 1, 1, 2: Applying the Pythagorean theorem, we have 1² + 1² = 2², which simplifies to 2 = 4. Thus, this set of side lengths does not represent a right triangle.
d) 7, 24, 25: Applying the Pythagorean theorem, we have 7² + 24² = 25², which simplifies to 49 + 576 = 625. Thus, this set of side lengths represents a right triangle.