Which of the following sets gives three possible side lengths of a valid triangle (all measurements in centimetres)?

Introduction / Context:
This question checks understanding of the triangle inequality, which tells us which sets of three segments can form a triangle. It is a core concept in geometry and appears frequently in aptitude tests to quickly eliminate impossible side length combinations.

Given Data / Assumptions:

• Each option lists three positive numbers representing possible side lengths of a triangle.
• All lengths are in centimetres.
• A valid triangle must satisfy the triangle inequality for every pair of sides.

Concept / Approach:
For three positive numbers a, b, c to form the sides of a triangle, all three triangle inequalities must hold simultaneously: a + b > c, b + c > a, and c + a > b. If even one of these fails or holds only as equality (a + b = c), the segments cannot form a non degenerate triangle. We check each option one by one using this rule and look for the set where all three inequalities are strictly satisfied.

Step-by-Step Solution:
Step 1: Option a: 2, 3, 6. Check 2 + 3 = 5, which is not greater than 6. So these cannot form a triangle. Step 2: Option b: 3, 4, 5. Check 3 + 4 = 7 > 5, 4 + 5 = 9 > 3, and 5 + 3 = 8 > 4. All three inequalities are satisfied, so this set can form a triangle. Step 3: Option c: 2.5, 3.5, 6. Here 2.5 + 3.5 = 6, which is equal to the third side, not greater. This would be a degenerate triangle, so it is not considered a valid triangle in most exam contexts. Step 4: Option d: 4, 4, 9. Check 4 + 4 = 8, which is less than 9, so these lengths also cannot form a triangle. Step 5: Therefore only option b gives a valid set of triangle side lengths.

Verification / Alternative Check:
The classic 3–4–5 set is well known as a Pythagorean triple, forming a right angled triangle. This supports our conclusion that 3 cm, 4 cm, and 5 cm can indeed form a triangle, and it is a useful example to remember for many geometry problems involving right triangles.

Why Other Options Are Wrong:
In option a, the sum of the two smaller sides is less than the largest side, so the segments cannot meet to close the triangle. In option c, the sum of the two smaller sides is exactly equal to the largest side, which would place all three points on a single straight line. In option d, again the two equal smaller sides sum to less than the largest side, violating the triangle inequality.

Common Pitfalls:
Students sometimes check only one inequality instead of all three, or they forget that equality is not enough for a proper triangle. Another mistake is to assume that if one pair of sides satisfies the condition, the others automatically do as well, which is not always true. Always check all three combinations carefully to be sure.

Final Answer:
The only possible set of triangle side lengths is 3 cm, 4 cm, 5 cm.