Question

If the points A(1,-2) B(2,3) C(a,2) and D(4,-3) form a parallelogram, find the value of a and hence find the height of the parallelogram taking AB as base?

Answer 1

we know midpoint of parellelogram's diagonal are always same.
here AC and BD are two diagonal of parellelogram
now midpoint of BD={(4+2)/2, (-3+3)/2}
(3,0)
hence midpoint of AC is also (3,0)
3=(1+a)/2
a=5
hence value of a=5

now,
AB=root {1+25} =root (26) {base
AC=root (32) {diagonal
BD=root (40) {diagonal

hence area of parellogram=1/2products of diagonals =height x base
1/2root (32) root (40)=height x root (26)
height=2root (2).root (40)/root (26)
=root (8 x 40/26)=root (320/26)