Final answer:
The set of lengths that makes up a right triangle is C) 14, 48, 52, as it fulfills the Pythagorean theorem which states that in a right triangle, the squares of the lengths of the two shorter sides should add up to the square of the length of the longest side, the hypotenuse.
Step-by-step explanation:
To determine which set of lengths makes up a right triangle, we can use the Pythagorean theorem. The theorem states that for a right triangle with legs a and b, and hypotenuse c, the relationship between the sides is a² + b² = c². Remember, the hypotenuse is the longest side in a right triangle.
Let's evaluate each set of lengths:
For A) 6, 13, 14: 6² + 13² = 36 + 169 = 205, but 14² = 196, which does not match, so this is not a right triangle.
For B) 5, 10, 20: 5² + 10² = 25 + 100 = 125, but 20² = 400, which again does not match, so this is not a right triangle.
For C) 14, 48, 52: 14² + 48² = 196 + 2304 = 2500, and 52² = 2704, so this matches, indicating that set C forms a right triangle.
For D) 18, 82, 82: Since the two sides are equal, this cannot be a right triangle as the hypotenuse must be longer than either of the legs.
Thus, the correct answer is C) 14, 48, 52.