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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.

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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 ‘a’ the largest side, ‘b’ be the second largest side, ‘c’ be the third largest side and ‘d’ 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,
  & a < b + c + d \cr
  & \Rightarrow a < 10 + 9 + 8 \cr
  & \Rightarrow a < 27 \cr} $$
The maximum value that ‘a’ can take is 26.
$\therefore $The maximum possible length of the given quadrilateral is 26.
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.

Last updated date: 28th Sep 2023
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