Question number 12Ans.Area 9, x+y=15, 7x-5y =15Area 9/2, x+y+9,7x-5y= -3I am find one equation of parallel side i.e. x+y=y but unable to find equations of non parallel side. And don't understand why given two answers.

ok so lets first understand why there shall be 2 answers. we can find the pt of intersection of x+y= 3 and 2x= y+3 as A(2, 1), which obviously is a vertex of the trapezium

now, diagonal intersection is given as P(3, 4). now consider the diagonal AP. if we name the trapezium as ABCD anti clckwise then diagonal AP shall continue to meet the trap at C. now, the ratio of diagonal is given as 2:1. so either AP:PC= 2:1 or CP:PA= 2:1. thats why there will be 2 possible answers. further, by properties of trapezium we know that CP/PA= BP/PD= 2:1 or 1:2 (2 cases that will have to be dealt separately). as you said you know the method to find parallel line’s eqn, so you know CB’s eqn and you can find coordinates of B by finding the intersection of this line with 2x= y+3. now, we have coordinates of B and P. we wanna find D, which obviously lies on x+y= 9, so its coordinates can be taken as (m, 3 – m). also, we know that BP/PD= 2:1 or 1:2 depending upon which case we are solving.

for BP/PD= 1:2, we can apply section formula to obtain D and hence the eqn of side CD. area is ez to find just divide the trap in 2 triangles and add the areas.