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.

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Solution

## Let the coordinates of $D$ is $\left(x,y\right)$. Since, $ABCD$ is a parallelogram. ∴ $AB=DC$ We have, $\stackrel{\to }{AB}=\stackrel{\to }{DC}\phantom{\rule{0ex}{0ex}}⇒3\stackrel{⏜}{i}-\left(-2\stackrel{⏜}{i}-\stackrel{⏜}{j}\right)=\left(\stackrel{⏜}{i}-2\stackrel{⏜}{j}\right)-\left(x\stackrel{⏜}{i}+y\stackrel{⏜}{j}\right)\phantom{\rule{0ex}{0ex}}⇒5\stackrel{⏜}{i}+\stackrel{⏜}{j}=\stackrel{⏜}{i}\left(1-x\right)+\stackrel{⏜}{j}\left(-2-y\right)\phantom{\rule{0ex}{0ex}}⇒1-x=5\mathrm{and}1=-2-y\phantom{\rule{0ex}{0ex}}⇒x=-4\mathrm{and}y=-3$ Hence, the coordinates of $D$ is $\left(-4,-3\right)$

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