Question

select the correct answer from each drop - down menu.
a quadrilateral has vertices ( a(11,-7), b(9,-4), c(11,-1) ), and ( d(13,-4) ).
quadrilateral ( abcd ) is a
if the vertex ( c(11,-1) ) were shifted to the point ( c^{prime}(11,1) ), quadrilateral ( a )

Explanation:

Step1: Calculate the slopes of the sides

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
For side \(AB\) with \(A(11,-7)\) and \(B(9,-4)\):
\(m_{AB}=\frac{-4 - (-7)}{9 - 11}=\frac{-4 + 7}{-2}=\frac{3}{-2}=-\frac{3}{2}\)
For side \(BC\) with \(B(9,-4)\) and \(C(11,-1)\):
\(m_{BC}=\frac{-1 - (-4)}{11 - 9}=\frac{-1 + 4}{2}=\frac{3}{2}\)
For side \(CD\) with \(C(11,-1)\) and \(D(13,-4)\):
\(m_{CD}=\frac{-4 - (-1)}{13 - 11}=\frac{-4 + 1}{2}=-\frac{3}{2}\)
For side \(DA\) with \(D(13,-4)\) and \(A(11,-7)\):
\(m_{DA}=\frac{-7 - (-4)}{11 - 13}=\frac{-7 + 4}{-2}=\frac{3}{2}\)

Step2: Analyze the slopes for parallelism

Since \(m_{AB}=m_{CD}=-\frac{3}{2}\) and \(m_{BC}=m_{DA}=\frac{3}{2}\), both pairs of opposite sides are parallel. So it is a parallelogram.
Now check the lengths of adjacent sides.
The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
Length of \(AB\): \(d_{AB}=\sqrt{(9 - 11)^2+(-4 + 7)^2}=\sqrt{(-2)^2+3^2}=\sqrt{4 + 9}=\sqrt{13}\)
Length of \(BC\): \(d_{BC}=\sqrt{(11 - 9)^2+(-1 + 4)^2}=\sqrt{2^2+3^2}=\sqrt{4 + 9}=\sqrt{13}\)
Check the product of slopes of adjacent sides (for perpendicularity). \(m_{AB}\times m_{BC}=-\frac{3}{2}\times\frac{3}{2}=-\frac{9}{4}
eq - 1\)

Answer:

parallelogram with non - perpendicular and non - congruent adjacent sides