Which of the following sets of three positive integers represents a Pythagorean triple, that is, the side lengths of a right angled triangle?

Introduction / Context:
This question checks understanding of Pythagorean triples, which are sets of three positive integers that can be the side lengths of a right angled triangle. Such triples satisfy the Pythagorean theorem, which is fundamental in geometry. The task is to look at each proposed triple and decide which one satisfies the condition a^2 + b^2 = c^2 for some ordering of the numbers where c is the largest.

Given Data / Assumptions:

• Each option contains three positive integers.
• We assume that if they form a right triangle, the largest of the three integers will be the hypotenuse.
• The Pythagorean theorem states that in a right triangle with legs a and b and hypotenuse c, we have a^2 + b^2 = c^2.
• Only one option is intended to be a valid Pythagorean triple.
• We must compute the squares and compare sums carefully.

Concept / Approach:
To identify a Pythagorean triple, first identify the largest number in the set as the potential hypotenuse. Then compute the squares of the other two numbers, add them and compare the result with the square of the largest number. If and only if a^2 + b^2 equals c^2, the triple is valid. We repeat this check for each option until we find one that satisfies the condition.

Step-by-Step Solution:
Step 1: For option a, the numbers are 68, 72 and 81, with 81 as the largest. Compute 68^2 = 4624 and 72^2 = 5184. The sum is 4624 + 5184 = 9808, while 81^2 = 6561, so 9808 is not equal to 6561. Option a is not a Pythagorean triple.Step 2: For option b, the numbers are 35, 38 and 42, with 42 as the largest. Compute 35^2 = 1225 and 38^2 = 1444. The sum is 2669, while 42^2 = 1764, so 2669 is not equal to 1764. Option b is not valid.Step 3: For option c, the numbers are 27, 38 and 42, with 42 again the largest. Compute 27^2 = 729 and 38^2 = 1444. The sum is 2173, but 42^2 is 1764, so there is no equality and option c is not a Pythagorean triple.Step 4: For option d, the numbers are 33, 44 and 55, with 55 as the largest. Compute 33^2 = 1089 and 44^2 = 1936. The sum is 1089 + 1936 = 3025. Now compute 55^2, which is also 3025. Since the sums match, option d is a Pythagorean triple.

Verification / Alternative check:
Once we have identified 33, 44 and 55 as a Pythagorean triple, we can note that this triple is actually a scaled version of the well known 3, 4, 5 triple. If we multiply 3, 4 and 5 by 11, we get 33, 44 and 55. Because scaling all sides of a right triangle by the same factor preserves the right angle, 33, 44 and 55 must also satisfy the Pythagorean relation. This provides a fast confirmation.

Why Other Options Are Wrong:
In each of the other options, when we square the two smaller numbers and add the results, the sum does not equal the square of the largest number. This means that they cannot be the sides of a right angled triangle. They might be sides of some triangle, but they do not satisfy the Pythagorean theorem, so they are not Pythagorean triples.

Common Pitfalls:
A common mistake is to assume that numbers that look close to each other or that differ by similar gaps automatically form a Pythagorean triple. Another pitfall is computing squares incorrectly or forgetting to identify the largest number as the candidate hypotenuse. Careful arithmetic and systematic checking prevent these errors.

Final Answer:
The only set that forms a Pythagorean triple is 33, 44, 55.