Is there any geometric demonstration that proves $m= \frac{y_2 -y_1}{x_2 - x_1}$ (the slope formula)?

Is there any geometric demonstration that proves $$m= \frac{y_2 -y_1}{x_2 - x_1}?$$

I know one algebraic demonstration. It's:

1) If you have a function $y=mx+b$ and two points $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$, you can make two equations and solve for $m$:

$$1) \: y_1=mx_1+b,$$ $$2) \: y_2=mx_2+b.$$

If you subtract equation $1$ from equation $2$, you get: $$y_2-y_1=mx_2+b-(mx_1+b)$$ $$y_2-y_1=mx_2+b-mx_1-b$$ $$y_2-y_1=mx_2-mx_1$$ $$y_2-y_1=m(x_2-x_1)$$ $$\frac{y_2 -y_1}{x_2 - x_1}=m.$$

That's the one I know, but since it's more of a geometry topic I'd rather like a geometric demonstration. Thanks.

I figured out this demonstration:

I am not an english speaker so I translated the best I could. Tell me what you think about it and if it´s valid. Thanks.

NOTE= In the second picture it should have said :"I´ll take a third point..."

• hmmm. I would say that slope is defined as $m = \frac {\text {rise}}{\text{run}} = \frac {y_1-y_0}{x_1-x_0}$ If we were looking at geometric descriptions we would be talking about the angle formed where the line intersects the coordinate axes. And that would be a different metric. – Doug M Jul 18 '17 at 20:22
• That´s true, but then my real question may be, how did mathematicians come up with $m= \frac{rise}{run}$? – Vmimi Jul 18 '17 at 20:27
• Mathematicians have defined it that way as it gives an idea of how steep the gradient is as well as its direction while being simplistic. The larger the magnitude of $m$, the steeper the slope is. The sign of $m$ indicates the direction. – Shuri2060 Jul 18 '17 at 20:30
• As for the math history, the algebraic linear equations are ancient, and are the geometric forms of a line both predate their representation on the coordinate plane by centuries. Credit Renee Descartes and Pierre de Fermat for this idea that the two fields could be unified. But, once the lines had been plotted, they would need a vocabulary to discuss these representations. – Doug M Jul 18 '17 at 20:38
• What do you mean by "slope", if not rise/run or $frac{y_1-y_0}{x_1-x_0}$? – Simply Beautiful Art Jul 18 '17 at 20:49

Just see that: $$m=\tan \alpha = \frac{y_b-y_a}{x_b-x_a}$$

• @jose esteban beleño barrozo: is it clear? – Arnaldo Jul 19 '17 at 15:50

To build off of what Doug M mentioned in his comment, slope is defined as the ratio of change in $y$ to the change in $x$.

Then we can see that the change between two points, or $\Delta p$, is given by their difference, $p_1 - p_0$. So the slope would be defined as,

$$m = \frac{\Delta y}{\Delta x} = \frac{y_1 - y_0}{x_1 - x_0}$$

Geometrically, the ratio of change in $y$ to change in $x$ is the "steepness". The rate at which both values change.

Slope of a line is a definition, but to show it is well defined (to show that no matter what points we use we will get the same slope) we may use topics of geometry. Specifically, triangle similarity.

Why did mathematicians decide that the slope was important?

It's important for a lot of reasons, but they mostly come down to differentiation. The wiki page has some beautiful geometrical demonstrations.