Final answer:
The side lengths of 9 mm, 12 mm, and 15 mm do form a Pythagorean triple since they satisfy the Pythagorean theorem. They are actually multiples of the smallest Pythagorean triple, which is 3, 4, 5.
Step-by-step explanation:
To determine whether the set of measures for the sides of a triangle is a Pythagorean triple, we apply the Pythagorean theorem which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Specifically, a² + b² = c².
Given the side lengths a = 9 mm, b = 12 mm, and c = 15 mm, we test the Pythagorean theorem:
- a² = 9² = 81
- b² = 12² = 144
- c² = 15² = 225
Adding the squares of 'a' and 'b,' we get:
Since the sum of the squares of 'a' and 'b' equals the square of 'c' (225 = 225), the given measures do form a Pythagorean triple.
However, these numbers are multiples of the smallest Pythagorean triple 3, 4, and 5. To find the original triple, we divide each side by their greatest common divisor, which in this case is 3:
- a' = 9 / 3 = 3
- b' = 12 / 3 = 4
- c' = 15 / 3 = 5
Therefore, the original Pythagorean triple is 3, 4, 5.