MYP Standard Mathematics Coordinate Geometry Basics Key Definitions

A grid formed by two perpendicular number lines, the x-axis (horizontal) and y-axis (vertical), used to locate points with ordered pairs $(x,y)$.

For points $A(x_1,y_1)$ and $B(x_2,y_2)$, the distance between them is
$$AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

For two distinct points $A(x_1,y_1)$ and $B(x_2,y_2)$ on a line, the gradient is
$$m=\frac{y_2-y_1}{x_2-x_1},\quad x_2\neq x_1$$

For endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$, the midpoint is
$$M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

Two non-vertical lines are parallel if they have the same gradient.

Two non-vertical, non-horizontal lines are perpendicular if their gradients multiply to $-1$. If one line has gradient $m\,(m\neq 0)$, then a perpendicular line has gradient $-\frac{1}{m}$.

A line through $(x_1,y_1)$ with gradient $m$ can be written as $$y-y_1=m(x-x_1)$$

The equation of a non-vertical line can be written as $$y=mx+c$$ where $m$ is the gradient and $c$ is the $y$-intercept.

A line can be written as $$ax+by+c=0$$ with constants $a,b,c$ (often integers, and sometimes with a required sign convention such as $a>0$).