Answer:
Yes, multiplying every length of a Pythagorean triple by the same whole number results in a Pythagorean triple
Step-by-step 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)
