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.
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.
Answers
GPT-4o mini
Leon is incorrect in his reasoning. The correct explanation is:
**No, the given side lengths can form a Pythagorean triple even if the lengths found in step 2 do not.**
To elaborate, although the side lengths 21, 28, and 35 were divided correctly by their GCF to obtain 3, 4, and 5 (which do form a Pythagorean triple), the original side lengths 21, 28, and 35 do not actually satisfy the Pythagorean theorem.
To check if the original lengths form a Pythagorean triple, we check whether:
\[
21^2 + 28^2 = 35^2
\]
Calculating the squares:
\[
21^2 = 441
\]
\[
28^2 = 784
\]
\[
35^2 = 1225
\]
Now adding \(21^2\) and \(28^2\):
\[
441 + 784 = 1225
\]
Indeed, \(1225 = 1225\), hence the original lengths **do** form a Pythagorean triple. Therefore, while Leon's approach of using the reduced lengths is valid in itself, he misstates the conclusion that the presence of a common factor alone means the original lengths must also be a Pythagorean triple is flawed reasoning and irrelevant in confirming their validity.
Leon’s solution is correct regarding the verification of the original lengths being a Pythagorean triple, but the broader statement he made about the common factors leading to such triples is misleading.
**No, the given side lengths can form a Pythagorean triple even if the lengths found in step 2 do not.**
To elaborate, although the side lengths 21, 28, and 35 were divided correctly by their GCF to obtain 3, 4, and 5 (which do form a Pythagorean triple), the original side lengths 21, 28, and 35 do not actually satisfy the Pythagorean theorem.
To check if the original lengths form a Pythagorean triple, we check whether:
\[
21^2 + 28^2 = 35^2
\]
Calculating the squares:
\[
21^2 = 441
\]
\[
28^2 = 784
\]
\[
35^2 = 1225
\]
Now adding \(21^2\) and \(28^2\):
\[
441 + 784 = 1225
\]
Indeed, \(1225 = 1225\), hence the original lengths **do** form a Pythagorean triple. Therefore, while Leon's approach of using the reduced lengths is valid in itself, he misstates the conclusion that the presence of a common factor alone means the original lengths must also be a Pythagorean triple is flawed reasoning and irrelevant in confirming their validity.
Leon’s solution is correct regarding the verification of the original lengths being a Pythagorean triple, but the broader statement he made about the common factors leading to such triples is misleading.