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: \\(3^2 + 4^2 = 9 + 16 = 25 = 5^2\\)

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.

Explanation:

Verify the relationship between the sets of lengths

$$
LATEXBLOCK0
$$

Scale the triple by the common factor

$$
LATEXBLOCK1
$$

Evaluate the statements

$$ k(a^2 + b^2) eq (ka)^2 + (kb)^2 \implies k^2(a^2 + b^2) = (ka)^2 + (kb)^2 = k^2 c^2 = (kc)^2 $$

Answer:

• (A) Yes, multiplying every length of a Pythagorean triple by the same whole number results in a Pythagorean triple. (Correct answer)
• (B) Yes, any set of lengths with a common factor is a Pythagorean triple.
• (C) No, the lengths of Pythagorean triples cannot have any common factors.
• (D) No, the given side lengths can form a Pythagorean triple even if the lengths found in step 2 do not.