Answer: Quadrilateral JKLM is a parallelogram.
Step-by-step explanation: Given that the co-ordinates of the vertices of a quadrilateral JKLM are J(0,3) , K(1,4) , L(2,1) and M(1,0).
We are to determine if the quadrilateral JKLM is a parallelogram or not.
The lengths of the sides of the quadrilateral JKLM can be calculated using distance formula as follows :
[tex]JK=\sqrt{(1-0)^2+(4-3)^2}=\sqrt{1+1}=\sqrt{2}~\textup{units},\\\\KL=\sqrt{(2-1)^2+(1-4)^2}=\sqrt{1+9}=\sqrt{10}~\textup{units},\\\\LM=\sqrt{(1-2)^2+(0-1)^2}=\sqrt{1+1}=\sqrt{2}~\textup{units},\\\\MJ=\sqrt{(0-1)^2+(3-0)^2}=\sqrt{1+9}=\sqrt{10}~\textup{units}.[/tex]
So, JK = LM and KL = MJ.
Also, the slopes of the sides are calculated as follows :
[tex]\textup{slope of JK},m_1=\dfrac{4-3}{1-0}=1,\\\\\\\textup{slope of KL},m_2=\dfrac{1-4}{2-1}=-3,\\\\\\\textup{slope of LM},m_3=\dfrac{0-1}{1-2}=1,\\\\\\\textup{slope of MJ},m_4=\dfrac{3-0}{0-1}=-3.[/tex]
So, slopes of the opposite sides are parallel. This implies that the opposite sides are equal parallel to each other.
Thus, quadrilateral JKLM is a parallelogram.
