# if vertices of paralleogram abcd are a(4,-1) b(5,3) c(2,5) d(1,1) if E is midpoint of AB find the coordinates of e and show that ar ABCD =2 arCDE

Dear Student,

We form our diagram from given information , As :

Here we assume coordinate of E ( x , y  )
We know formula for coordinate of mid point ( x , y ) =

As E is mid point of AB , x1 = 4, x2 =  5 and  y1 = - 1 , y2 = 3  , ( And we assume coordinate of E (x,y ) ), So

So, Coordinate of E ( 4.5 , 1 )                                                ( Ans )

We know area of triangle from given three points  :

Area  =

To find area of triangle ABD , Here x1 = 4 , x2 =  5 , x3 = 1  and  y1 = - 1 , y2 =  3 , y3 = 1

So,

Area of triangle ABD  = = 7 unit square

We know diagonal of parallelogram divide it in two equal parts , So

Area of parallelogram ABCD = 2 ( Area of triangle ABD ) , So

Area of parallelogram ABCD = 2 ( 7 ) = 14 square unit

And

To find area of triangle CDE , Here x1 = 2 , x2 =  1 , x3 = 4.5  and  y1 = 5 , y2 =  1 , y3 = 1

So,

Area of triangle CDE  = = 7 unit square

Now from area of parallelogram ABCD and triangle CDE we can say :

Area of ABCD  = 2 Area of triangle CDE                                                             ( Hence proved )