Question

consider the graph of quadrilateral abcd.
what is the most specific name for quadrilateral abcd?
rectangle
parallelogram
rhombus
square

Explanation:

Step1: Find the lengths of sides

First, we use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) to find the lengths of the sides of quadrilateral \(ABCD\).

For \(AB\): \(A(-2,3)\), \(B(2,2)\)
\(d_{AB}=\sqrt{(2 - (-2))^2+(2 - 3)^2}=\sqrt{4^2+(- 1)^2}=\sqrt{16 + 1}=\sqrt{17}\)

For \(BC\): \(B(2,2)\), \(C(0,-3)\)
\(d_{BC}=\sqrt{(0 - 2)^2+(-3 - 2)^2}=\sqrt{(-2)^2+(-5)^2}=\sqrt{4 + 25}=\sqrt{29}\)

For \(CD\): \(C(0,-3)\), \(D(-4,-2)\)
\(d_{CD}=\sqrt{(-4 - 0)^2+(-2 - (-3))^2}=\sqrt{(-4)^2+1^2}=\sqrt{16 + 1}=\sqrt{17}\)

For \(DA\): \(D(-4,-2)\), \(A(-2,3)\)
\(d_{DA}=\sqrt{(-2 - (-4))^2+(3 - (-2))^2}=\sqrt{2^2+5^2}=\sqrt{4 + 25}=\sqrt{29}\)

So, \(AB = CD=\sqrt{17}\) and \(BC = DA=\sqrt{29}\), which means opposite sides are equal.

Step2: Check slopes for parallelism

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\)

Slope of \(AB\): \(m_{AB}=\frac{2 - 3}{2 - (-2)}=\frac{-1}{4}=-\frac{1}{4}\)

Slope of \(BC\): \(m_{BC}=\frac{-3 - 2}{0 - 2}=\frac{-5}{-2}=\frac{5}{2}\)

Slope of \(CD\): \(m_{CD}=\frac{-2 - (-3)}{-4 - 0}=\frac{1}{-4}=-\frac{1}{4}\)

Slope of \(DA\): \(m_{DA}=\frac{3 - (-2)}{-2 - (-4)}=\frac{5}{2}\)

Since \(m_{AB}=m_{CD}=-\frac{1}{4}\) and \(m_{BC}=m_{DA}=\frac{5}{2}\), opposite sides are parallel. So, \(ABCD\) is a parallelogram.

Now, check if it is a rectangle: For a rectangle, adjacent sides should be perpendicular (product of slopes should be \(- 1\)).

\(m_{AB}\times m_{BC}=-\frac{1}{4}\times\frac{5}{2}=-\frac{5}{8}
eq - 1\), so not a rectangle.

Check if it is a rhombus: For a rhombus, all sides should be equal. But \(AB=\sqrt{17}\) and \(BC=\sqrt{29}\), so sides are not equal, so not a rhombus (and hence not a square as square is a special rhombus and rectangle).

Answer:

parallelogram