Question

How many differently shaped triangles exist in which no two sides are of the same length, each side is of integral unit length and the perimeter of the triangle is less than 14 units?

• A. 3
• B. 4
• C. 5
• D. 6

Hint:

List scalene integer-sided triangles a less than b less than c with a + b greater than c and perimeter under 14; check systematically starting from the smallest side.

Answer 1

The Correct Option is C

A different way to count these triangles is to fix the perimeter first (an integer less than 14, so at most 13) and see how many scalene triangles with distinct integer sides fit each perimeter.

1. Perimeter up to 8: the smallest possible scalene triangle with distinct integer sides is (2,3,4), with perimeter 9. So no triangle has perimeter 8 or less.
2. Perimeter 9: only (2,3,4) works, since $2+3>4$ holds and all sides differ.
3. Perimeter 10: checking combinations like (2,3,5), $2+3=5$ is not greater than 5, so it fails, and no other distinct-integer combination for 10 satisfies the strict inequality.
4. Perimeter 11: (2,4,5) works since $2+4=6>5$. No other combination for 11 satisfies all the conditions.
5. Perimeter 12: (3,4,5) works since $3+4=7>5$. Other combinations like (2,4,6) fail the inequality, so only one triangle here.
6. Perimeter 13: both (2,5,6) and (3,4,6) work, since $2+5=7>6$ and $3+4=7>6$. No other combination for 13 satisfies all the conditions.

Adding these up: 1 (perimeter 9) + 0 (perimeter 10) + 1 (perimeter 11) + 1 (perimeter 12) + 2 (perimeter 13) = 5 triangles in total.

Let's summarize:

• Going perimeter by perimeter instead of side by side gives the same five triangles: (2,3,4), (2,4,5), (3,4,5), (2,5,6), (3,4,6).
• The triangle inequality $a + b > c$ is the deciding filter at every perimeter value.

So the answer is 5.