Question

A(6,1) , B(8,2) and C(9,4) are the vertices of a parallelogram ABCD. If E is the midpoint of DC, find the area of ΔADE 

Answer 1

Let  (x,y) be the coordinates of D and ( x ',y')be thee coordinates of E. since, the diagonals of a parallelogram bisect each other at the same point, therefore

`(x+8)/2=(6+9)/2⇒x =7`

`(y+2)/2 = (1+4)/2 ⇒ y=3`

Thus, the coordinates of D are ( 7,3)

E is the midpoint of DC, therefore

`x' = (7+9)/2 ⇒ x' = 8`

`y' =(3+4)/2 ⇒ y' = 7/2 `

Thus, the coordinates of E are `(8,7/2)`

`"let" A (x_1, y_1) = A(6,1) ,E (x_2,y_2) = E (8,7/2) and D(x_3,y_3) = D(7,3) Now`

`"Area" (Δ ABC) = 1/2 [ x_1(y_2-y_3)+x_2(y_3-y_1) +x_3(y_1-y_2)]`

`=1/2[6(7/2-3)+8(3-1)+7(1-7/2)]`

`=1/2[3/2]`

`=3/4` sq. units 

Hence, the area of the triangle ΔADE  is `3/ 4`sq. units