As Long as Possible
Weekly Problem 40 - 2013
Given three sides of a quadrilateral, what is the longest that the fourth side can be?
Given three sides of a quadrilateral, what is the longest that the fourth side can be?
Age
11 to 14
Challenge level
Problem
The length of each side of a quadrilateral $ABCD$ is a whole number of centimetres. Given that $AB=4 \; \text{cm}$, $BC = 5 \; \text{cm}$ and $CD = 6 \; \text{cm}$, what is the maximum possible length of the fourth side $DA$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
$14 \; \text{cm}$. The length of $AD$ must be less than $15 \; \text{cm}$, since $15 \; \text{cm}$ would be its length if all four points lay in a straight line. However, by making angles $ABC$ and $BCD$ close to $180^{\circ}$, $AD$ can be made close to $15 \; \text{cm}$ in length.
As the length of $AD$ is a whole number of centimetres, its maximum value, therefore is $14\; \text{cm}$