If the three sides of a triangle are 12, 22 and m units, then which of the following is correct?

The correct answer is 10 < m < 34.

This problem is based on the Triangle Inequality Theorem, which states that for any valid triangle with sides a, b, and c, the sum of any two sides must be strictly greater than the third side. Equality is never allowed — if the sum of two sides equals the third, the triangle degenerates into a straight line (zero area), which is not a valid triangle.

The three sides given are 12, 22, and m. We must apply the Triangle Inequality Theorem to all three combinations of sides.

Condition 1: The sum of the two known sides must be greater than m:

12+22>m
34>m
This gives us: m<34

Condition 2: The sum of m and the smaller known side must be greater than the larger known side:

m+12>22
m>22-12
m>10

Condition 3: The sum of m and the larger known side must be greater than the smaller known side:

m+22>12
m>-10
Since m is a length, it is always positive, so this condition is automatically satisfied and adds no new restriction.

Combining the binding conditions from Condition 1 and Condition 2:

10<m<34

Why not ≤ (less than or equal to)?
Notice the inequality is strict on both sides. If m = 10, then the sides 12, 22, and 10 would fail the test: 10 + 12 = 22, which is not greater than 22 — the three points would be collinear, forming a degenerate triangle. Similarly, if m = 34, then 12 + 22 = 34, again a degenerate case. Both endpoints are excluded.

Therefore, the correct range for m is 10 < m < 34.