Question

Quadrilateral JKLM has vertices J(0,3) , K(1,4) , L(2,1) and M(1,0). Determine if quadrilateral JKLM is a parallelogram

Answer 1

JKLM is a parallelogram

Explanation:

A parallelogram is a quadrilateral with two pair of parallel sides. Its four sides has not to be congruent between them and its angles has not to be right.

From the figure attached it can be seen that sides JM and KL are parallel, and sides JK and ML are parallel too.

Side JK is
`√(1^2+1^2) = √(2)` units length

Side ML is
`√(1^2+1^2) = √(2)` units length

Side JM is
`√(1^2+3^2) = √(10)` units length

Side KL is
`√(1^2+3^2) = √(10)` units length

Therefore, the four sides are not congruent.

Side JL is
`√(2^2+2^2) = √(8)` units length

Side KM is 4 units length

Since the two diagonals (sides JL and KM) are not equal, then the four internal angles of the quadrilateral are not equal between them; in consequence they are not right angles.

Answer 2

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 :

`JK=√((1-0)^2+(4-3)^2)=√(1+1)=√(2)~\textup{units},\\\\KL=√((2-1)^2+(1-4)^2)=√(1+9)=√(10)~\textup{units},\\\\LM=√((1-2)^2+(0-1)^2)=√(1+1)=√(2)~\textup{units},\\\\MJ=√((0-1)^2+(3-0)^2)=√(1+9)=√(10)~\textup{units}.`

So, JK = LM and KL = MJ.

Also, the slopes of the sides are calculated as follows :

`\textup{slope of JK},m_1=(4-3)/(1-0)=1,\\\\\\\textup{slope of KL},m_2=(1-4)/(2-1)=-3,\\\\\\\textup{slope of LM},m_3=(0-1)/(1-2)=1,\\\\\\\textup{slope of MJ},m_4=(3-0)/(0-1)=-3.`

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.