Which of the following sets of three side lengths cannot represent the sides of a right angled triangle?

Introduction / Context:
This is a geometry and Pythagoras theorem question. In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Many common sets of integers that satisfy this relationship are called Pythagorean triples. The question asks us to identify which of the given sets of three side lengths does not satisfy the relationship and therefore cannot form a right angled triangle.

Given Data / Assumptions:
- Four sets of side lengths are given: (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 15, 18).
- All lengths are positive numbers representing the sides of a potential triangle.
- We must check whether each set can satisfy Pythagoras theorem in some ordering.

Concept / Approach:
For any given triple of side lengths, we first identify the largest side and treat it as the possible hypotenuse. Then we compute the square of that side and compare it with the sum of the squares of the other two sides. If the equality holds, the triple can represent a right angled triangle. If it fails, the triple is not a Pythagorean triple and cannot represent a right angled triangle with those side lengths.

Step-by-Step Solution:
Step 1: For 3, 4, 5 the largest side is 5. Check 3^2 + 4^2 = 9 + 16 = 25 and 5^2 = 25, so it is a right angled triangle. Step 2: For 5, 12, 13 the largest side is 13. Check 5^2 + 12^2 = 25 + 144 = 169 and 13^2 = 169, so it is a right angled triangle. Step 3: For 8, 15, 17 the largest side is 17. Check 8^2 + 15^2 = 64 + 225 = 289 and 17^2 = 289, so it is also a right angled triangle. Step 4: For 12, 15, 18 the largest side is 18. Compute 12^2 + 15^2 = 144 + 225 = 369. Step 5: Compute 18^2 = 324. Step 6: Since 369 is not equal to 324, 12, 15 and 18 do not satisfy Pythagoras theorem and therefore cannot form a right angled triangle.

Verification / Alternative check:
We can also verify by checking if any side ordering for 12, 15, 18 works, but the hypotenuse must always be the largest side. Swapping labels among the smaller sides does not change the sum of their squares, so 144 + 225 remains 369 regardless of order. Since the square of 18 is fixed as 324, no rearrangement will satisfy the equality. This confirms that 12, 15, 18 cannot represent a right angled triangle.

Why Other Options Are Wrong:
3, 4, 5 is the most well known Pythagorean triple. 5, 12, 13 and 8, 15, 17 are also standard triples used frequently in geometry and aptitude problems. All of these satisfy a^2 + b^2 = c^2 for suitable ordering of sides. Only 12, 15, 18 fails that condition and is therefore the only incorrect candidate for a right angled triangle.

Common Pitfalls:
A common error is to miscalculate one of the squares or the sum, for example computing 15^2 incorrectly. Another mistake is to assume any triple that looks similar to a known Pythagorean triple must also work without checking the arithmetic. Always square the sides carefully and verify the equality rather than relying purely on pattern recognition.

Final Answer:
The set of lengths that cannot be the sides of a right angled triangle is 12, 15, 18.