Triangle Inequality Math Example 3

Follow the full solution, then compare it with the other examples linked below.

easy

Which of these sets of side lengths cannot form a triangle? (a) 3, 4, 5. (b) 1, 2, 3. (c) 5, 8, 12.

1
Step 1: Check (a): $3 + 4 = 7 > 5$ ✓ — valid triangle.
2
Step 2: Check (b): $1 + 2 = 3$ , which is NOT greater than 3 (it equals 3) ✗ — not a valid triangle (degenerate).
3
Step 3: Check (c): $5 + 8 = 13 > 12$ ✓ — valid triangle.

Answer

Set (b) cannot form a triangle.

For set (b), the sum of the two smaller sides equals the largest side (

1 + 2 = 3

), which makes the 'triangle' degenerate — the three points are collinear. The triangle inequality requires a strict inequality (

>

, not

\ge

), so equal sums are not allowed.

About Triangle Inequality

The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

Can sides of length 4, 7, and 10 form a triangle? Check all three triangle inequality conditions.

Two sides of a triangle have lengths 8 and 13. Find the range of possible lengths for the third side

In [formula], [formula] and [formula]. The perimeter is 24. Is this a valid triangle? Find [formula]