Coordinate Geometry

Length and midpoint

Length of a line

If you have two points $A(x_1, y_1)$ and $B(x_2, y_2)$, then the length of the line segment $AB$ is found using the distance formula:

$\text{Length} = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Midpoint

To find the middle point between $A(x_1, y_1)$ and $B(x_2, y_2)$, use this formula:

$\text{Midpoint} = \left(\frac{1 + 4}{2},\ \frac{2 + 6}{2}\right) = (2.5,\ 4)$

Equations of linear graphs

Finding the equation of a line

If you know the gradient and a point on the line $(x_1, y_1)$, use this formula:

This is called the point-gradient form.

$y - 2 = 3\left(x - (-1)\right) \quad\Rightarrow\quad y - 2 = 3x + 3 \quad\Rightarrow\quad y = 3x + 5$

Getting information from an equation

You can find the gradient and $y$-intercept straight from the equation.

Rearrange it to the $y = mx + c$ form:

$4y = -5x + 8 \quad\Rightarrow\quad y = -\frac{5}{4}x + 2$

The gradient is $-\dfrac{5}{4}$ and the $y$-intercept is 2.

Parallel and perpendicular lines

Parallel lines have the same gradient.

• The gradient of the given line is 4.
• Use the point $(1, -3)$ and plug into the point-gradient form:

$y + 3 = 4(x - 1) \quad\Rightarrow\quad y + 3 = 4x - 4 \quad\Rightarrow\quad y = 4x - 7$

Perpendicular lines are lines that meet at a right angle ($90^\circ$), like the corners of a square. They have opposite reciprocal gradients. If a line has gradient $m$, then a perpendicular line has gradient:

First, write it in $y = mx + c$ form: $y = \frac{3}{2}x + \frac{1}{2}$. The gradient is $\frac{3}{2}$, so the perpendicular gradient is $-\frac{2}{3}$ (reciprocal and change sign).

Perpendicular bisector

A perpendicular bisector cuts the line in half (through the midpoint) at a right angle.

Step 1: find the midpoint.

$\left(\frac{-3 + 9}{2},\ \frac{8 + (-2)}{2}\right) = (3,\ 3)$

Step 2: find the gradient of $AB$.

$m = \frac{-2 - 8}{9 - (-3)} = \frac{-10}{12} = -\frac{5}{6}$

Step 3: find the perpendicular gradient. Flip and change the sign: $\dfrac{6}{5}$.

Step 4: plug the midpoint into the point-gradient form.

$y - 3 = \frac{6}{5}(x - 3) \quad\Rightarrow\quad y = \frac{6}{5}x - \frac{18}{5} + 3 \quad\Rightarrow\quad y = \frac{6}{5}x - \frac{3}{5}$