Question

Is KLMN a parallelogram

Answer 1

Given data:

The given vertices of the parallelogram are K(2, 7), L(6, 12), M(13, 13) and N(9, 8).

In parallelogram opposite sides are equal and parallel.

`\begin{gathered} KL=\sqrt[]{(6-2)^2+(12-7)^2} \\ =\sqrt[]{4^2+5^2} \\ =\sqrt[]{16+25} \\ =\sqrt[]{41} \end{gathered}`

The slope of KL is,

`\begin{gathered} m=(12-7)/(6-2) \\ =(5)/(4) \end{gathered}`

The measuremment of the side MN is,

`\begin{gathered} MN=\sqrt[]{(9-13)^2+(8-13)^2} \\ =\sqrt[]{16+25} \\ =\sqrt[]{41} \end{gathered}`

The slope of the MN is,

`\begin{gathered} m^(\prime)=(8-13)/(9-13) \\ =(-5)/(-4) \\ =(5)/(4) \end{gathered}`

The LM length is,

`\begin{gathered} LM=\sqrt[]{(13-6)^2+(13-12)^2} \\ =\sqrt[]{49+1} \\ =\sqrt[]{50} \end{gathered}`

The slope of LM is,

`\begin{gathered} m_1=(13-12)/(13-6) \\ =(1)/(7) \end{gathered}`

The length NK is,

`\begin{gathered} NK=\sqrt[]{(2-9)^2+(7-8)^2} \\ =\sqrt[]{49+1} \\ =\sqrt[]{50} \end{gathered}`

The slope of NK is,

`\begin{gathered} m_2=(7-8)/(2-9) \\ =(-1)/(-7) \\ =(1)/(7) \end{gathered}`

As, the opposite sides are equal and parallel, so the given quadrilaterl KLMN is parallelogram.

Thus, Yes it is parallelogram, so first option is correct.