Which of the following sets of three side lengths cannot form a triangle (all measurements in centimetres)?

Introduction / Context:
This problem is another application of the triangle inequality, focusing on identifying which set of three positive numbers cannot be side lengths of a triangle. This concept is fundamental in geometry and often appears in objective tests to quickly eliminate impossible configurations.

Given Data / Assumptions:

• Each option lists three positive numbers representing potential side lengths.
• All side lengths are in centimetres.
• A valid triangle must satisfy the triangle inequality: sum of any two sides must be strictly greater than the third side.

Concept / Approach:
For three lengths a, b, and c to form a non degenerate triangle, it must be true that a + b > c, b + c > a, and c + a > b. If even one of these is violated (less than or equal to), a triangle cannot be formed. We check each option to see where the inequality fails. The option where the sum of the two shorter sides does not exceed the longest side is the one that cannot form a triangle.

Step-by-Step Solution:
Step 1: Option a: 5, 6, 7. Check 5 + 6 = 11 > 7, 6 + 7 = 13 > 5, 7 + 5 = 12 > 6. All conditions are satisfied, so this set can form a triangle. Step 2: Option b: 5, 8, 15. Order them as 5, 8, 15 with 15 the largest. Check 5 + 8 = 13, which is less than 15. This fails the triangle inequality. Step 3: Because 5 + 8 is not greater than 15, these three lengths cannot form a triangle. Step 4: Option c: 8, 15, 18. Check 8 + 15 = 23 > 18, 15 + 18 = 33 > 8, 18 + 8 = 26 > 15. All good, so this set can form a triangle. Step 5: Option d: 6, 7, 11. Check 6 + 7 = 13 > 11, 7 + 11 = 18 > 6, 11 + 6 = 17 > 7. This set can also form a triangle.

Verification / Alternative Check:
Once you find a single failing case in a set (like 5 + 8 ≤ 15 in option b), you know that the set cannot form a triangle. For the other options, all three inequalities hold strictly, so there is no contradiction. Thus, option b is uniquely the impossible combination.

Why Other Options Are Wrong:
Option a corresponds to a typical acute or obtuse triangle configuration with lengths that clearly satisfy the inequalities. Options c and d also show sums of the two smaller sides that comfortably exceed the third side. Only option b fails the basic requirement that the two shorter sides together must exceed the longest side.

Common Pitfalls:
Sometimes students check only the smallest two sides against the largest and forget to verify the other two inequalities. While in many cases that is enough, relying on only one inequality can occasionally lead to mistakes. Another common error is misordering the sides, which can confuse which pair should be compared with the largest side. Sorting the three numbers first and then checking the sum of the two smaller ones against the largest is a safe and quick strategy.

Final Answer:
The set of side lengths that cannot form a triangle is 5 cm, 8 cm, 15 cm.