How do you prove algebraically, that the following vertices are a rectangle. I can plot it on a graph, but I can't figure out how to prove using algebra.
Help.
1 way to do it:
Let the given points correspond to $\displaystyle A,B,C,\text{ and }D$ respectively.
Then to show that $\displaystyle ABCD$ is a rectangle, we must show that each interior angle measures $\displaystyle 90^{\circ}$ . To do this, we must need to only compare the slopes of line segments $\displaystyle AB$ with $\displaystyle BC$, and $\displaystyle BC$ with $\displaystyle CD$ and etc...
Do you get what I'm saying?
OK.
Perpindicular lines will have slopes that are the negative recirocal respective to each other.
So $\displaystyle AB$ has slope $\displaystyle m_1=\frac{5-0}{-1-1}=-\frac{5}{2}$
and $\displaystyle BC$ has slope $\displaystyle m_2=\frac{7-5}{4-(-1)}=\frac{2}{5}$
Now, since $\displaystyle -\frac{1}{m_1}=m_2$ it follows that $\displaystyle AB$ is perpindicular to $\displaystyle BC$.