Question

ABCD is a parallelogram. If the coordinates of A, B, C are (−2, −1), (3, 0) and (1, −2) respectively, find the coordinates of D.

Answer 1

Let the coordinates of $D$ is $\left(x,y\right)$.
Since, $ABCD$ is a parallelogram.
∴ $AB=DC$
We have,

$$\begin{gathered} \overrightarrow{AB}=\overrightarrow{DC} \\ \Rightarrow 3\stackrel{⏜}{i}-\left(-2\stackrel{⏜}{i}-\stackrel{⏜}{j}\right)=\left(\stackrel{⏜}{i}-2\stackrel{⏜}{j}\right)-\left(x\stackrel{⏜}{i}+y\stackrel{⏜}{j}\right) \\ \Rightarrow 5\stackrel{⏜}{i}+\stackrel{⏜}{j}=\stackrel{⏜}{i}\left(1-x\right)+\stackrel{⏜}{j}\left(-2-y\right) \\ \Rightarrow 1-x=5\mathrm{and}1=-2-y \\ \Rightarrow x=-4\mathrm{and}y=-3 \end{gathered}$$
Hence, the coordinates of $D$ is $\left(-4,-3\right)$