Final answer:
Only Option D offers a set of lengths that form a right triangle similar to the original triangle with lengths of 5, 12, and 13. The triangle with sides 15, 36, and 39 is proportional to the given triangle by a factor of 3, which makes it similar according to the Pythagorean theorem.
Step-by-step explanation:
A triangle with side lengths of 5, 12, and 13 is a right triangle, according to the Pythagorean theorem, which 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 (a² + b² = c²).
When looking for a triangle that is similar to the given triangle, we need to find a set of sides that have the same proportional relationships. This is because similar triangles have all the same angles and their sides are proportional. The given triangle satisfies the Pythagorean theorem (5² + 12² = 13²).
Let's test the sets of sides given in the options:
Option A (8, 13, 14): 8² + 13² is not equal to 14², so it is not a right triangle and cannot be similar.
Option B (10, 24, 26): This is a typographical mistake and should be disregarded.
Option C (9, 15, 25): 9² + 15² is not equal to 25², so it is not a right triangle and cannot be similar.
Option D (15, 36, 39): 15² + 36² equals 39², which means this is also a right triangle and the side lengths are proportional to the original triangle (by a factor of 3). Therefore, this set of sides represents a triangle that is similar to the original.
Therefore, the correct answer is Option D: A similar triangle could have sides with lengths of 15, 36, and 39.