Which of these pairs of numbers are two out of three integer-valued side lengths of a right triangle? Select all that are true. (Hint: for positive integers a,b,c, and k, ka, kb, and kc are side lengths of a right triangle if and only if a, b, and care side lengths of a right triangle.) 15,18 15, 30 20,52 8,10 18, 27

Answer: To determine which pair of numbers could be sides of a right triangle, we need to see if they could be consistent with the Pythagorean theorem which states that for a right triangle with sides a and b, and the hypotenuse (the longest side) c, the following relationship holds true: a² + b² = c².

Let's analyze each pair:

1. 15, 18: Let's assume 15 and 18 are the two shorter sides of the triangle; to find the hypotenuse, we use the Pythagorean theorem. 15² + 18² = 225 + 324 = 549 We can't find an integer that, when squared, equals 549. So, this pair doesn't form a right triangle.

2. 15, 30: Again, assuming these are the two shorter sides and applying the theorem. 15² + 30² = 225 + 900 = 1125 No integer squared is 1125, so it is not a right triangle.

3. 20, 52: Assuming these are the two shorter sides. 20² + 52² = 400 + 2704 = 3104 The square root of 3104 is 56, which is an integer. Therefore, 20, 52 could be the two shorter sides of a right triangle.

4. 8, 10: Assuming these are the two shorter sides. 8² + 10² = 64 + 100 = 164 No integer squared gives 164, so this is not the side of a right triangle.

5. 18, 27: Assuming these are the two shorter sides. 18² + 27² = 324 + 729 = 1053 No integer squared gives 1053, so it is not a right triangle.

From the pairs given, only the pair (20, 52) could be part of a right triangle as integer-valued side lengths, with the hypotenuse being 56.

Extra: The question hints at the idea that if you have a set of numbers, a, b, and c, which satisfy the Pythagorean theorem (a right triangle), then if you multiply all those three numbers by the same positive integer k, the resulting numbers, ka, kb, and kc will also form the side lengths of a right triangle. This is because when you multiply a, b, and c all by the same constant k, the relationship a² + b² = c² scales to (ka)² + (kb)² = (kc)². This is possible because (ka)² = k²*a², (kb)² = k²*b², and (kc)² = k²*c², thus maintaining the necessary proportion for the Pythagorean theorem to hold true. This is a fundamental property of similar triangles - all their corresponding sides are in the same ratio (in this case, the ratio created by the constant k).