Which of the following could be the side lengths of a right triangle: 3, 13, and 14; 4, 5, and 6; 5, 12, and 13; 5, 10, and 15?

To determine which set of numbers can be the side lengths of a right triangle, we can use the Pythagorean theorem. According to this theorem, in a right triangle, the squares of the lengths of the two shorter sides (legs) add up to the square of the length of the longest side (hypotenuse).

Let’s analyze each option:

• 3, 13, and 14:
  
  Applying the theorem: 3² + 13² = 9 + 169 = 178
  
  14² = 196.
  
  Since 178 ≠ 196, this set does not form a right triangle.
• 4, 5, and 6:
  
  Applying the theorem: 4² + 5² = 16 + 25 = 41
  
  6² = 36.
  
  Since 41 ≠ 36, this set does not form a right triangle.
• 5, 12, and 13:
  
  Applying the theorem: 5² + 12² = 25 + 144 = 169
  
  13² = 169.
  
  Since 169 = 169, this set forms a right triangle.
• 5, 10, and 15:
  
  Applying the theorem: 5² + 10² = 25 + 100 = 125
  
  15² = 225.
  
  Since 125 ≠ 225, this set does not form a right triangle.

In conclusion, the only side lengths that could form a right triangle from the options given are 5, 12, and 13.