Question

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the coordinates of the vertices of quadrilateral $jklm$ are $j(-3,2),k(3,5),l(9, - 1)$, and $m(2,-3)$.
which statement correctly describes whether quadrilateral $jklm$ is a rhombus?
quadrilateral $jklm$ is a rhombus because opposite sides are parallel and all four sides have the same length.
quadrilateral $jklm$ is not a rhombus because there are no pairs of parallel sides.
quadrilateral $jklm$ is not a rhombus because there is only one pair of opposite sides that are parallel.
quadrilateral $jklm$ is not a rhombus because opposite sides are parallel but the four sides do not all have the same length.

Explanation:

Step1: Recall slope - formula

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate slope of $JK$

For points $J(-3,2)$ and $K(3,5)$, $m_{JK}=\frac{5 - 2}{3-(-3)}=\frac{3}{6}=\frac{1}{2}$.

Step3: Calculate slope of $KL$

For points $K(3,5)$ and $L(9,-1)$, $m_{KL}=\frac{-1 - 5}{9 - 3}=\frac{-6}{6}=-1$.

Step4: Calculate slope of $LM$

For points $L(9,-1)$ and $M(2,-3)$, $m_{LM}=\frac{-3+1}{2 - 9}=\frac{-2}{-7}=\frac{2}{7}$.

Step5: Calculate slope of $MJ$

For points $M(2,-3)$ and $J(-3,2)$, $m_{MJ}=\frac{2 + 3}{-3 - 2}=\frac{5}{-5}=-1$.
Since $m_{KL}=m_{MJ}=-1$ and $m_{JK}
eq m_{LM}$, there is one pair of parallel sides.
Now, recall the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step6: Calculate length of $JK$

$d_{JK}=\sqrt{(3 + 3)^2+(5 - 2)^2}=\sqrt{36 + 9}=\sqrt{45}=3\sqrt{5}$.

Step7: Calculate length of $KL$

$d_{KL}=\sqrt{(9 - 3)^2+(-1 - 5)^2}=\sqrt{36+36}=\sqrt{72}=6\sqrt{2}$.
Since the side - lengths are not equal and there is only one pair of parallel sides.

Answer:

Quadrilateral $JKLM$ is not a rhombus because there is only one pair of opposite sides that are parallel.