# Section 2.1

## Change, Distance, Midpoint
Let $(x_1,y_1)$ and $(x_2,y_2)$ be any two points on the $xy$-plane. The following image will be used:

![Graph!](images/graph.png)

* The change in $x$ is $\Delta x = x_2 - x_1$.
* The change in $y$ is $\Delta y = y_2 - y_1$.
* The distance between the two points is $d=\sqrt{(\Delta x)^2+(\Delta y)^2}$.
* The midpoint, $M$, is $\left( \frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

:::{prf:definition} Collinear
:label: collinear
Given three points $A$, $B$, and $C$. Let $d(A,B)$ denote the distance point $A$ is away from point $B$ (using the distance formula).

If $d(A,B)\le d(B,C) \le d(A,C)$ and $d(A,B)+d(B,C) = d(A,C)$, then we say $A$, $B$, and $C$ are **collinear**. Otherwise, we say the points are **not collinear**.