1. ## Analytic Geometry Q5

Question:
Show the points $\displaystyle A(-1,0)$ , $\displaystyle B(5,2)$ , $\displaystyle C(8,7)$ & $\displaystyle D(2,5)$ 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

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

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

Slope of line $\displaystyle AB$:

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

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

Slope of line $\displaystyle CD$:

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

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

Slope of line $\displaystyle BC$:

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

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

Slope of line $\displaystyle AD$:

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

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

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