Question

the coordinates of the vertices of quadrilateral (abcd) are (a(-6,3)), (b(-1,5)), (c(3,1)), and (d(-2,-2)).
which statement correctly describes whether quadrilateral (abcd) is a rhombus?
(\bigcirc) quadrilateral (abcd) is not a rhombus because there is only one pair of opposite sides that are parallel
(\bigcirc) quadrilateral (abcd) is not a rhombus because opposite sides are parallel but the four sides do not all have the same length
(\bigcirc) quadrilateral (abcd) is a rhombus because opposite sides are parallel and all four sides have the same length
(\bigcirc) quadrilateral (abcd) is not a rhombus because there are no pairs of parallel sides

Explanation:

Step1: Calculate the slope of each side

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
For side \(AB\) with \(A(-6,3)\) and \(B(-1,5)\):
\(m_{AB}=\frac{5 - 3}{-1-(-6)}=\frac{2}{5}\)
For side \(BC\) with \(B(-1,5)\) and \(C(3,1)\):
\(m_{BC}=\frac{1 - 5}{3-(-1)}=\frac{-4}{4}=- 1\)
For side \(CD\) with \(C(3,1)\) and \(D(-2,-2)\):
\(m_{CD}=\frac{-2 - 1}{-2 - 3}=\frac{-3}{-5}=\frac{3}{5}\)
For side \(DA\) with \(D(-2,-2)\) and \(A(-6,3)\):
\(m_{DA}=\frac{3-(-2)}{-6-(-2)}=\frac{5}{-4}=-\frac{5}{4}\)

Step2: Calculate the length of each side

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
For \(AB\):
\(d_{AB}=\sqrt{(-1 + 6)^2+(5 - 3)^2}=\sqrt{25 + 4}=\sqrt{29}\)
For \(BC\):
\(d_{BC}=\sqrt{(3 + 1)^2+(1 - 5)^2}=\sqrt{16+16}=\sqrt{32}\)
For \(CD\):
\(d_{CD}=\sqrt{(-2 - 3)^2+(-2 - 1)^2}=\sqrt{25 + 9}=\sqrt{34}\)
For \(DA\):
\(d_{DA}=\sqrt{(-6 + 2)^2+(3 + 2)^2}=\sqrt{16 + 25}=\sqrt{41}\)

Answer:

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