If a triangle and a parallelogram are drawn on the same base and between the same parallels, then the area of the triangle is equal to half the area of the parallelogram and Triangles drawn on the same base and between the same parallel lines are equal in area ABCD is a trapezium in which  AB∥CD and  DC  =  40cm and  AB  =  60cm. If  X and  Y are respectively the mid-points of  AD and  BC, ar(trap .DCYX)  =  p × ar (trap. XYBA). Find the value of p.

The problem is asking for the ratio of the areas of two trapezoids, DCYX and XYBA, which share the same height and have bases that are parallel.

Step 1: Understand the problem The trapezoids DCYX and XYBA share the same height (the distance between the parallel lines AB and CD). The bases of the trapezoids are the segments DC and AB for trapezoid DCYX, and XY and BA for trapezoid XYBA.

Step 2: Apply the formula for the area of a trapezoid The area of a trapezoid is given by the formula 1/2*(sum of the lengths of the parallel sides)*height.

Step 3: Calculate the lengths of the bases of the trapezoids The length of DC is given as 40 cm, and the length of AB is given as 60 cm. Since X and Y are the midpoints of AD and BC respectively, the length of XY is the average of the lengths of AB and DC, which is (60+40)/2 = 50 cm.

Step 4: Substitute the lengths of the bases into the formula for the area of a trapezoid The area of trapezoid DCYX is 1/2*(40+50)height = 45height square cm. The area of trapezoid XYBA is 1/2*(50+60)height = 55height square cm.

Step 5: Find the ratio of the areas of the trapezoids The ratio of the areas of trapezoids DCYX and XYBA is (45height)/(55height) = 45/55 = 9/11.

So, the value of p is 9/11.