The sides of a Quadrilateral are in A.P. The semi perimeter of the Quadrilateral is 40. The second largest side of the Quadrilateral is three times the smallest side. Find the largest side.

To solve this problem, we need to use the properties of an arithmetic progression (AP) and the formula for the semi-perimeter of a quadrilateral.

Step 1: Define the sides of the quadrilateral Since the sides are in AP, we can denote them as a-d, a, a+d, a+2d, where a is the second term and d is the common difference.

Step 2: Use the given conditions to form equations We know that the semi-perimeter is 40, so the sum of all sides is 80. Therefore, we have the equation: (a-d) + a + (a+d) + (a+2d) = 80 Solving this equation gives us: 4a + 2d = 80

We also know that the second largest side (a+d) is three times the smallest side (a-d). Therefore, we have the equation: a + d = 3(a - d) Solving this equation gives us: 2d = 2a

Step 3: Solve the equations We can solve these two equations simultaneously to find the values of a and d. From the second equation, we get a = d. Substituting this into the first equation gives us: 4a + 2a = 80 6a = 80 a = 80/6 = 13.33

Substituting a = 13.33 into the equation a = d gives us d = 13.33

Step 4: Find the largest side The largest side of the quadrilateral is a + 2d = 13.33 + 2*13.33 = 40

So, the largest side of the quadrilateral is 40 units.