Question

ABCD is a parallelogram whose vertices are (-3,2), (1,5) and (2,-4) respectively, then the coordinates of fourth vertex is

Answer 1

let A(-3,2) B(1,5), C(2,-4) and D(x,y)
parrallel lines have same slope
slope of AB=Slope of CD
$\frac{5-2}{1-(-3)}= \frac{y-(-4)}{x-2}$
solve it 3x-4y-22=0....(1)
slope of AD=slope of BC
$\frac{y-2}{x-(-3)}= \frac{5-(-4)}{1-2}$
solve it 9x+y+25=0....(2)
solve 
(1) and (2) we get x=-2 ,y=-7
the fourth vertex is (-2,-7)

Answer 2

Coordinates of the midpoint of AC = Coordinates of the midpoint of BD.
A(-3,2), B(1,5), C(2,-4) and D(x,y)
((-3+2)/2,(2-4)/2) = ((1+x)/2,(5+y)/2)
(-1/2,-2/2) = ((x+1)/2,(y+5)/2)
(-1/2,-1) = ((x+1)/2,(y+5)/2)
(x+1)/2 = -1/2, (y+5)/2 = -1
Therefore x = -2 and y =-7

The fourth vertices is (-2,-7)