Question

# A quadrilateral has distinct integer side lengths. If the second largest side has length 10, then find the maximum possible length of the largest side.A.25B.26C.27D.28

Hint: We are going to solve the given problem by using the idea that in a quadrilateral, the largest side length should be less than the sum of lengths of the remaining three sides.

Let â€˜aâ€™ the largest side, â€˜bâ€™ be the second largest side, â€˜câ€™ be the third largest side and â€˜dâ€™ be the smallest side of the given quadrilateral.
$\Rightarrow$a > b > c > d
Given that the length of the second largest side b = 10.
Then the largest possible values for d and c are 8 and 9 respectively.
By the triangle inequality,
\eqalign{ & a < b + c + d \cr & \Rightarrow a < 10 + 9 + 8 \cr & \Rightarrow a < 27 \cr}
The maximum value that â€˜aâ€™ can take is 26.
$\therefore$The maximum possible length of the given quadrilateral is 26.
Note:
Triangular inequality states that the sum of any two sides of a triangle must be greater than the measure of the third side. In the same way it is applicable to quadrilaterals.