Question

plot and connect the points in the order listed below. when you are done, choose the word best identifies the resulting shape/polygon.

$a(-1,-5), b(1,0), c(7,0), d(5,-5)$

plot all of the points above to make a polygon.
you may click a plotted point to delete it.

Explanation:

Step1: Recall the distance formula

The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate the lengths of the sides

For side $AB$ with $A(-1,-5)$ and $B(1,0)$:
$d_{AB}=\sqrt{(1 - (-1))^2+(0 - (-5))^2}=\sqrt{(2)^2+(5)^2}=\sqrt{4 + 25}=\sqrt{29}$
For side $BC$ with $B(1,0)$ and $C(7,0)$:
$d_{BC}=\sqrt{(7 - 1)^2+(0 - 0)^2}=\sqrt{(6)^2+0^2}=6$
For side $CD$ with $C(7,0)$ and $D(5,-5)$:
$d_{CD}=\sqrt{(5 - 7)^2+(-5 - 0)^2}=\sqrt{(-2)^2+(-5)^2}=\sqrt{4 + 25}=\sqrt{29}$
For side $DA$ with $D(5,-5)$ and $A(-1,-5)$:
$d_{DA}=\sqrt{(-1 - 5)^2+(-5-(-5))^2}=\sqrt{(-6)^2+0^2}=6$

Step3: Analyze the slopes of the sides

The slope of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
Slope of $AB$: $m_{AB}=\frac{0-(-5)}{1 - (-1)}=\frac{5}{2}$
Slope of $BC$: $m_{BC}=\frac{0 - 0}{7 - 1}=0$
Slope of $CD$: $m_{CD}=\frac{-5 - 0}{5 - 7}=\frac{-5}{-2}=\frac{5}{2}$
Slope of $DA$: $m_{DA}=\frac{-5-(-5)}{-1 - 5}=0$
Since $AB = CD=\sqrt{29}$, $BC=DA = 6$ and $m_{AB}=m_{CD}$, $m_{BC}=m_{DA}$, the opposite sides are equal and parallel.

Answer:

Parallelogram