Number of Triangles with Integral Sides

CAT 2000 Slot 1 · Quantitative Ability · Medium · Geometry

This is a medium Quantitative Ability question from the CAT 2000 Slot 1 paper. It tests Geometry. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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What is the number of distinct triangles with integral valued sides and perimeter 14?

A.
6
B.
5
C.
4
D.
3

Answer

Explanation

Let sides be $a \le b \le c$ with $a + b + c = 14$ . By triangle inequality, $c < \frac{14}{2} = 7 \implies c \le 6$ . Possible integer tuples $(a, b, c)$ satisfying $a + b > c$ :

$(6, 6, 2)$
$(6, 5, 3)$
$(6, 4, 4)$
$(5, 5, 4)$ Total distinct triangles = 4.

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Number of Triangles with Integral Sides

Explanation

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