Straight Lines and Coordinate Geometry

Starting point: Years 9 and 10

Gradient

The gradient $m$ tells you how steep a line is. It is the rise divided by the run.

$$m=\frac{\text{rise}}{\text{run}}=\frac{y_2-y_1}{x_2-x_1}$$

• Positive gradient: the line goes up to the right.
• Negative gradient: the line goes down to the right.
• Horizontal line: $m=0$. Vertical line: gradient is undefined.

The equation of a line

$$y=mx+c$$

$m$ is the gradient and $c$ is the y-intercept, where the line crosses the y-axis. In the taxi story $m=2$ and $c=4$.

Distance and midpoint

For points $(x_1,y_1)$ and $(x_2,y_2)$:

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

The distance formula is just Pythagoras on the grid.

Parallel and perpendicular lines

A line perpendicular to $y=3x-2$ has gradient $-\frac{1}{3}$.

General form

Lines are also written as $ax+by+c=0$. To find the gradient, rearrange into $y=mx+c$. For $2x+3y-6=0$: $3y=-2x+6$, so $y=-\frac{2}{3}x+2$.

Quick summary

• Gradient $m=\frac{y_2-y_1}{x_2-x_1}$: rise over run.
• $y=mx+c$: $m$ is the gradient, $c$ is the y-intercept.
• Distance uses Pythagoras. The midpoint is the average of the coordinates.
• Parallel lines: same gradient. Perpendicular lines: $m_1{m}_2=-1$.
• Point-gradient form: $y-y_1=m(x-x_1)$.

Quiz