Question

 ABCD is a parallelogram with vertices  \[A ( x_1 , y_1 ), B \left( x_2 , y_2 \right), C ( x_3 , y_3 )\]   . Find the coordinates  of the fourth vertex D in terms of  \[x_1 , x_2 , x_3 , y_1 , y_2 \text{ and }  y_3\]

   

Answer 1

Suppose the coordinates of D be (x, y).
Since diagonals of a parallelogram bisect each other.

Therefore the midpoint of AC is the midpoint of BD, i.e

\[\left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right) \text{ and }       \left( \frac{x_2 + x}{2}, \frac{y_2 + y}{2} \right)\]  respectively.

\[\Rightarrow x_1 + x_3 = x_2 + x \text{ and } y_1 + y_3 = y_2 + y\]

\[ \Rightarrow x = x_1 + x_3 - x_2 \text{ and } y = y_1 + y_3 - y_2 \]

\[\text{ Thus coordinates of D are } \left( x_1 + x_3 - x_2 , y_1 + y_3 - y_2 \right)\]