# If A (-2, -1), B (k, 0), C (4, P) and (1, 2) are the vertices of a parallelogram find the value of k and p.â€‹

We are given the coordinates,

A(-2, -1), B(k, 0), C(4, P), D(1, 2)

In a parallelogram, diagonals bisect each other.

So, Midpoint of AC = Midpoint of BD

$\\ \text{Using midpoint formula,}\\ \Rightarrow\left(\frac{-2+4}{2}, \frac{-1+p}{2}\right)=\left(\frac{k+1}{2}, \frac{0+2}{2}\right) \\ \Rightarrow\left(\frac{2}{2}, \frac{-1+p}{2}\right)=\left(\frac{k+1}{2}, \frac{2}{2}\right)\\ \\ {(\text { equating x coordinates })}\\ {\Rightarrow \frac{2}{2}=\frac{k+1}{2}} \\ \Rightarrow 2=k+1 \\ \Rightarrow 2-1=k \\ \Rightarrow k=1 \\ {(\text { equating y coordinates })}\\ {\Rightarrow \frac{2}{2}=\frac{-1+p}{2}}\\ \Rightarrow 2=-1+\mathrm{p} \\ \Rightarrow 3=p \\ \therefore p=3$

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