Question

If $\left(3,-4\right)$ and $\left(-6,5\right)$ are the extremities of the diagonals of a parallelogram and $\left(-2,1\right)$ is its third vertex, then its fourth vertex is

• A. (-1,0)
• B. (-1,1)
• C. (0,-1)
• D. None of these

Answer 1

The correct option is A

(-1,0)

Step 1- Write the midpoint formula.

Let $A(3,-4),C(-6,5)$ be the endpoints of the diagonals of a parallelogram and $B(-2,1)$ is its third vertex.

Let $D(x,y)$ be the fourth vertex.

Since the diagonals of a parallelogram bisect each other thus the midpoints of $AC$ and $BD$will be equal.

The midpoint formula is $(x,y)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$, where $(x_1,y_1),(x_2,y_2)$ are the coordinates of the vertices.

Step 2- Find the fourth vertex.

$\begin{array}{rcl}\mathrm{midpoint}\mathrm{of}AC& =& \mathrm{midpoint}\mathrm{of}BD\\ \left(\frac{3-6}{2},\frac{-4+5}{2}\right)& =& \left(\frac{-2+x}{2},\frac{1+y}{2}\right)\\ \left(-\frac{3}{2},\frac{1}{2}\right)& =& \left(\frac{-2+x}{2},\frac{1+y}{2}\right)\\ & \Rightarrow & \frac{-2+x}{2}=-\frac{3}{2}\\ & \Rightarrow & -2+x=-3\\ & \Rightarrow & x=-1\\ & \Rightarrow & \frac{1+y}{2}=\frac{1}{2}\\ & \Rightarrow & y=0\end{array}$

Thus, the fourth vertex is $(-1,0)$.

Hence, option A is correct.