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the coordinates of the vertices of quadrilateral ( a b c d ) are ( a(-5,1), b(-2,5), c(5,3) ), and ( d(2,-1) ).
drag and drop the choices into each box to correctly complete the sentences.
the slope of ( overline{a b} ) is, the slope of ( overline{b c} ) is, the slope of ( overline{c d} ) is, and the slope of ( overline{a d} ) is
quadrilateral ( a b c d ) is because
( -\frac{2}{7} quad -\frac{1}{3} quad \frac{3}{2} quad \frac{4}{3} ) a parallelogram a trapezoid neither a parallelogram nor a trapezoid
both pairs of opposite sides are parallel only one pair of opposite sides is parallel
neither pair of opposite sides is parallel

Explanation:

Step1: Calculate the slope of \(\overline{AB}\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(A(-5,1)\) and \(B(-2,5)\), \(m_{AB}=\frac{5 - 1}{-2-(-5)}=\frac{4}{3}\)

Step2: Calculate the slope of \(\overline{BC}\)

For points \(B(-2,5)\) and \(C(5,3)\), \(m_{BC}=\frac{3 - 5}{5-(-2)}=-\frac{2}{7}\)

Step3: Calculate the slope of \(\overline{CD}\)

For points \(C(5,3)\) and \(D(2,-1)\), \(m_{CD}=\frac{-1 - 3}{2 - 5}=\frac{-4}{-3}=\frac{4}{3}\)

Step4: Calculate the slope of \(\overline{AD}\)

For points \(A(-5,1)\) and \(D(2,-1)\), \(m_{AD}=\frac{-1 - 1}{2-(-5)}=-\frac{2}{7}\)

Step5: Determine the type of quadrilateral

Since \(m_{AB}=m_{CD}=\frac{4}{3}\) and \(m_{BC}=m_{AD}=-\frac{2}{7}\), both pairs of opposite sides are parallel.

Answer:

The slope of \(\overline{AB}\) is \(\frac{4}{3}\), the slope of \(\overline{BC}\) is \(-\frac{2}{7}\), the slope of \(\overline{CD}\) is \(\frac{4}{3}\), and the slope of \(\overline{AD}\) is \(-\frac{2}{7}\). Quadrilateral \(ABCD\) is a parallelogram because both pairs of opposite sides are parallel.