Question

Leon verified that the side lengths 21, 28, 35 form a Pythagorean triple using this procedure. Step 1: Find the greatest common factor of the given lengths: 7 Step 2: Divide the given lengths by the greatest common factor: 3, 4, 5 Step 3: Verify that the lengths found in step 2 form a Pythagorean triple: 3 squared + 4 squared = 9 + 16 = 25 = 5 squared Leon states that 21, 28, 35 is a Pythagorean triple because the lengths found in step 2 form a Pythagorean triple. Which explains whether or not Leon is correct? Yes, multiplying every length of a Pythagorean triple by the same whole number results in a Pythagorean triple. Yes, any set of lengths with a common factor is a Pythagorean triple. No, the lengths of Pythagorean triples cannot have any common factors. No, the given side lengths can form a Pythagorean triple even if the lengths found in step 2 do not.

Answer 1

Yes, multiplying every length of a Pythagorean triple by the same whole number results in a Pythagorean triple

Explanation:

The given side lengths are;

21, 28, and 35

From the given side lengths, the greatest common factor = 7

Therefore, we have;

3 × 7, 4 × 7, and 5 × 7

A Pythagorean triple is formed by three numbers, 'a', 'b', and, 'c', which are both positive and integers, that are related as follows;

c² = a² + b²

Therefore, for the three numbers, we get;

3, 4, and 5 is a Pythagorean triple

5² = 3² + 4²

Multiplying both sides by 7² gives;

(5)² × (7)² = (3)² × (7)² + (4)² × (7)²

By the laws of indices, (a·b)ⁿ = aⁿ × bⁿ

Therefore;

(5 × 7)² = (3 × 7)² + (4 × 7)²

Therefore, multiplying every length of the Pythagorean triple by the same whole number results in a Pythagorean triple.

(The relationship can also be shown using similar triangles, as multiplying the side lengths of a triangle by the same whole number, gives a similar triangle, having the same relationship between its sides)