Possible Integer Side Lengths of Quadrilateral

CAT 2022 Slot 3 · QA · Easy · Geometry

All CAT 2022 Slot 3 questionsMore Geometry questionsAll CAT QA

The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side is

  1. A.

    6

  2. B.

    4

  3. C.

    5

  4. D.

    3

Answer

C

Explanation

By Generalized Polygon Inequality, the length of any side of a polygon must be strictly less than the sum of the lengths of all other sides.

Let the fourth side be xx (an integer).

  1. x<1+2+4  ⟹  x<7x < 1 + 2 + 4 \implies x < 7
  2. 4<1+2+x  ⟹  x>14 < 1 + 2 + x \implies x > 1

Combining these conditions: 1<x<71 < x < 7

Since xx is an integer, the possible values of xx are: x∈{2,3,4,5,6}x \in \{2, 3, 4, 5, 6\} There are 5 possible integer lengths.

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