Question

If $A(1,2),B(4,3),C(6,6)$ are the vertices of a parallelogram $ABCD,$ then find the coordinate of the forth vertex.

Answer 1

Let $D(x,y)$ be the coordinate of the fourth vertex of the parallelogram $ABCD$

We know that diagonals bisect each other in a parallelogram.

$\Rightarrow$Mid-point of $AC=$Mid-point of $BD$

Using mid-point formula,we have $(x,y)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

$\Rightarrow (\frac{1+6}{2},\frac{2+6}{2})=(\frac{x+4}{2},\frac{y+3}{2})$

$\Rightarrow \frac{x+4}{2}=\frac{7}{2},\frac{y+3}{2}=\frac{8}{2}$

$\Rightarrow x+4=7,y+3=8$

$\Rightarrow x=7-4=3,y=8-3=5$

$\therefore$ the fourth vertex is $(3,5)$