Number of Triangles with Integral Sides

CAT 2000 Slot 1 · QA · Medium · Geometry

All CAT 2000 Slot 1 questionsMore Geometry questionsAll CAT QA

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

  1. A.

    6

  2. B.

    5

  3. C.

    4

  4. D.

    3

Answer

C

Explanation

Let sides be a≤b≤ca \le b \le c with a+b+c=14a + b + c = 14. By triangle inequality, c<142=7  ⟹  c≤6c < \frac{14}{2} = 7 \implies c \le 6. Possible integer tuples (a,b,c)(a, b, c) satisfying a+b>ca + b > c:

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

Practise this under exam conditions

Sign in to solve it with a live timer, the on-screen CAT calculator, and streak and accuracy tracking across every question you attempt.

Solve in the workspace