CAT 2022 Slot 3 · Quant · Question 22

A quadrilateral closes up only if no single side is as long as the other three combined.

That gives one condition for each side, but only two of them bite here.

Let the fourth side be x centimetres.

First, x itself must be shorter than the other three together: x<1+2+4=7.

Since x is a whole number, that means x\le 6.

Second, the longest known side, 4, must be shorter than the rest together: 4<1+2+x.

That gives 4<3+x, so x>1 and therefore x\ge 2.

The conditions involving the sides 1 and 2 are automatically satisfied, since those are the smallest.

So x can be 2, 3, 4, 5 or 6.

That is 5 possible lengths — the third option.

Shortcut. Only the longest side's condition and the new side's own condition matter, so the answer is the whole numbers strictly between 1 and 7. Counting those gives 6-2+1=5 without checking each side in turn.