Coordinate Geometry

Within a Cartesian Coordinate System, each point in space has an 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 and a 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 representing its horizontal position and vertical position. The coordinates are written as an ordered pair, (𝑥, 𝑦).

1. Distance Formula

The length of a line joining two points can be found based on the Pythagorean Theorem.

$AB=\sqrt{(x_2-x_1 )^2+(y_2-y_1 )^2}$

2. Midpoint Formula

The midpoint of a line joining two points,  is given by:

$M=[\frac{1}{2} (x_2-x_1 ),\frac{1}{2} (y_2-y_1 )]$

3. Slope of a Line

The slope of a line is the measure of its gradient at any given point.  could stand for “modulus of slope.” – the reason is unknown*

$Slope,m^*=\frac{Δy}{Δx}$

Positive Slope $(m>0)$

$\alpha$ is an acute angle with the positive sense of the $x-axis$.
$Gradient = tan\theta$

Negative Slope $(m<0)$

$\beta$ is an obtuse angle with the positive sense of the $x-axis$.
$Gradient = tan\theta$

In both cases, the gradient of the line passing through $A(x_1,y_1 )$ and $B(x_2,y_2)$ is given by: $\frac{Δy}{Δx}=tan\theta$

4. Parallel Lines

If $l_1$ and $l_2$ are parallel lines, $l_1$ and $l_2$ must be equally inclined to the positive sense of the $x-axis$. Hence, $tan\theta$ is the slope of $l_1$ and $l_2$.

5. Perpendicular Lines

If $l_1$ and $l_2$ are perpendicular their slopes must be negative reciprocals of each other:

$m_1=-\frac{1}{m_2}$ OR $m_1 m_2=-1$