1. ## Analytic Geometry Q5

Question:
Show the points $A(-1,0)$ , $B(5,2)$ , $C(8,7)$ & $D(2,5)$ are vertices of a parallelogram.

I don't what I'm suppose to find the prove that it's a parallelogram. Should I find the slope m of the lines? or should I find the distance between the points? and what does vertices mean?!

EDIT: This question is related to the lesson of Inclination and Slope of a line.

2. Yes, you have to find the slope of the lines and see that

$m_{AB}=m_{CD}$ and $m_{BC}=m_{AD}$

3. $A(-1,0)$ , $B(5,2)$ , $C(8,7)$ & $D(2,5)$

Slope of line $AB$:

$A(-1,0)$ & $B(5,2)$

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2-0}{5-(-1)} = \frac{1}{3}$

Slope of line $CD$:

$C(8,7)$ & $D(2,5)$

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5-7}{2-8} = \frac{1}{3}$

Slope of line $BC$:

$B(5,2)$ & $C(8,7)$

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7-2}{8-5} = \frac{5}{3}$

Slope of line $AD$:

$A(-1,0)$ & $D(2,5)$

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5-0}{2-(-1)} = \frac{5}{3}$

4. If $m_{AB}=m_{CD}\Rightarrow AB\parallel CD$
and if $m_{BC}=m_{AD}\Rightarrow BC\parallel AD$