Question

what is the best name for quadrilateral abcd given the points a(-3,3), b(1,7), c(4,4), and d(0,0)?
rhombus
rectangle
parallelogram
square

Explanation:

Step1: Calculate the slopes of the sides

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
For \(AB\): \(m_{AB}=\frac{7 - 3}{1-(-3)}=\frac{4}{4} = 1\).
For \(BC\): \(m_{BC}=\frac{4 - 7}{4 - 1}=\frac{-3}{3}=-1\).
For \(CD\): \(m_{CD}=\frac{0 - 4}{0 - 4}=\frac{-4}{-4}=1\).
For \(DA\): \(m_{DA}=\frac{3 - 0}{-3 - 0}=\frac{3}{-3}=-1\).

Step2: Check the properties of parallelogram

In a parallelogram, opposite sides are parallel.
Since \(m_{AB}=m_{CD}=1\) and \(m_{BC}=m_{DA}=-1\), \(AB\parallel CD\) and \(BC\parallel DA\).

Step3: Check for rectangle (perpendicular adjacent sides)

For adjacent sides to be perpendicular, the product of their slopes should be \(- 1\).
\(m_{AB}\times m_{BC}=1\times(-1)=-1\), \(m_{BC}\times m_{CD}=(-1)\times1=-1\), \(m_{CD}\times m_{DA}=1\times(-1)=-1\), \(m_{DA}\times m_{AB}=(-1)\times1=-1\).

Step4: Calculate the lengths of the sides

The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
\(AB=\sqrt{(1 + 3)^2+(7 - 3)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\).
\(BC=\sqrt{(4 - 1)^2+(4 - 7)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\).
Since \(AB
eq BC\), it is not a rhombus or a square.

Answer:

rectangle