Question

Points $A\left(4,3\right),B\left(6,4\right),C\left(5,-6\right)\mathrm{and}D\left(-3,5\right)$ are the vertices of a parallelogram. State whether the following statement is true or false. Justify your answer.

Answer 1

Determine whether the given statement is true or false.

Find that whether the given vertices form a parallelogram or not by determining the distance between the two vertices.

Apply the distance formula: $D=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Find the distance between $A\left(4,3\right)\mathrm{and}B\left(6,4\right)$.

$\begin{array}{rcl}{D}_{AB}& =& \sqrt{{\left(6-4\right)}^2+{\left(4-3\right)}^2}\\ & \Rightarrow & D_{AB}=\sqrt{4+1}\\ & \Rightarrow & D_{AB}=\sqrt{5}\end{array}$

Find the distance between $B\left(6,4\right)\mathrm{and}C\left(5,-6\right)$.

$\begin{array}{rcl}{D}_{BC}& =& \sqrt{{\left(5-6\right)}^2+{\left(-6-4\right)}^2}\\ & \Rightarrow & D_{BC}=\sqrt{1+100}\\ & \Rightarrow & D_{BC}=\sqrt{101}\end{array}$

Find the distance between $C\left(5,-6\right)\mathrm{and}D\left(-3,5\right)$.

$\begin{array}{rcl}{D}_{CD}& =& \sqrt{{\left(-3-5\right)}^2+{\left(5-\left(-6\right)\right)}^2}\\ & \Rightarrow & D_{CD}=\sqrt{64+121}\\ & \Rightarrow & D_{CD}=\sqrt{185}\end{array}$

Find the distance between $A\left(4,3\right)\mathrm{and}D\left(-3,5\right)$.

$\begin{array}{rcl}{D}_{AD}& =& \sqrt{{\left(-3-4\right)}^2+{\left(5-3\right)}^2}\\ & \Rightarrow & D_{AD}=\sqrt{49+4}\\ & \Rightarrow & D_{AD}=\sqrt{53}\end{array}$

Since the distances are different, the given vertices cannot form a parallelogram.

Hence, the given statement is false.