Question

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

Explanation:

Step1: Calculate the slopes of the sides

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
For \(JK\) with \(J(-3,2)\) and \(K(3,5)\):
\(m_{JK}=\frac{5 - 2}{3-(-3)}=\frac{3}{6}=\frac{1}{2}\)
For \(KL\) with \(K(3,5)\) and \(L(9,-1)\):
\(m_{KL}=\frac{-1 - 5}{9 - 3}=\frac{-6}{6}=-1\)
For \(LM\) with \(L(9,-1)\) and \(M(2,-3)\):
\(m_{LM}=\frac{-3-(-1)}{2 - 9}=\frac{-2}{-7}=\frac{2}{7}\)
For \(MJ\) with \(M(2,-3)\) and \(J(-3,2)\):
\(m_{MJ}=\frac{2-(-3)}{-3 - 2}=\frac{5}{-5}=-1\)
Since \(m_{JK}
eq m_{LM}\) and \(m_{KL}=m_{MJ}=-1\), there is only one pair of parallel sides.

Step2: Calculate the lengths of the sides

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(JK\):
\(d_{JK}=\sqrt{(3 + 3)^2+(5 - 2)^2}=\sqrt{36 + 9}=\sqrt{45}=3\sqrt{5}\)
For \(KL\):
\(d_{KL}=\sqrt{(9 - 3)^2+(-1 - 5)^2}=\sqrt{36+36}=\sqrt{72}=6\sqrt{2}\)
For \(LM\):
\(d_{LM}=\sqrt{(2 - 9)^2+(-3 + 1)^2}=\sqrt{49 + 4}=\sqrt{53}\)
For \(MJ\):
\(d_{MJ}=\sqrt{(-3 - 2)^2+(2 + 3)^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2}\)
The sides are not all equal.

Answer:

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