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Show that the point $A(0,0)$, $B(2,5)$, $C(7,-3)$, $D(9,2)$ are vertices of the parallelogram and using determinants to find its area.

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closed as off-topic by JonMark Perry, Claude Leibovici, mrp, Namaste, TheGeekGreek Feb 8 '17 at 14:08

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  • $\begingroup$ If you plot the points on a Cartesian plane and find the slope of the opposite sides to be equal, meaning that the lines are parallel, then you can prove it is a parallelogram. $\endgroup$ – Harnoor Lal Feb 8 '17 at 9:03
  • $\begingroup$ Then create a matrix whose columns are vectors representing the two non-parallel sides and compute its determinant. $\endgroup$ – SquirtleSquad Feb 8 '17 at 9:23
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Note that $A$, $B$, $C$, $D$ are not in correct order.

Since $A+D=B+C=(9,2)$, $AD$ and $BC$ bisect each other.

Hence, $ABDC$ is a parallelogram.

$$\text{Area of }ABDC = \left| \vec{AB} \times \vec{AC} \right|$$

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