CAT 2022 Slot 3 QA Question 7: 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

Question

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

Accepted Answer

For a quadrilateral to be possible, the longest side must be less than the sum of the other three sides.
Let the fourth side be $n$ cm.
If 4 is the longest side, then
$1 + 2 + n > 4$
$n > 1$
If $n$ is the longest side, then
$n 
$n 
So $n$ can be any integer from 2 to 6.
$n = 2, 3, 4, 5, 6$
There are