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CAT 2022 Slot 3 — QA Question 10

Basics of QuadrilateralsEasy

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

Answer & solution

  • 5

  • B

    4

  • C

    3

  • D

    6

Solution

Easy

Generalise the triangle inequality to a quadrilateral: any one side must be less than the sum of the other three, and greater than the differences they can produce. With sides 1,2,41,2,4 and unknown xx, find the integer range for xx.

1

Quadrilateral side rule. Each side lies strictly between the sum of the other three and the largest "shortfall". The binding bounds here:

x<1+2+4=7(less than sum of the others)x>4−(1+2)=1(must be able to close the figure)\begin{aligned} &x<1+2+4=7 \quad\text{(less than sum of the others)}\\ &x>4-(1+2)=1 \quad\text{(must be able to close the figure)} \end{aligned}
2

Count integers in $1 x∈{2,3,4,5,6}⇒5 valuesx\in\{2,3,4,5,6\}\Rightarrow 5\text{ values}

Possible lengths of the fourth side=5\text{Possible lengths of the fourth side}=\mathbf{5}

The lower bound: with sides 1,2,41,2,4, the longest reachable "gap" the fourth side must bridge means xx must exceed 4−1−2=14-1-2=1; otherwise the four segments can't form a closed quadrilateral.

CAT 2022 Slot 3 QA Q10: The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, — Solution | TheCATExam