So, today I was observing a class that I will be a TA for this semester and the professor started to talk about the distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Well, my mind wandered a little and I started to think about slope. That's when I noticed, with a little bit of algebra we can convert the distance formula into a representation of slope.
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$d^2=(x_2-x_1)^2+(y_2-y_1)^2$
$\left(\frac{d}{x_2-x_1}\right)^2=1+\left(\frac{y_2-y_1}{x_2-x_1}\right)^2$ $\frac{y_2-y_1}{x_2-x_1}=\sqrt{\left(\frac{d}{x_2-x_1}\right)^2-1}$
I was wondering if anyone knows of any practical reason to use this, or if it's utterly pointless. My first impression is that it's pointless, unless you are given distance and two $x$ values and asked to find slope. But excluding that very unlikely case, I cannot think of a reason.