What you need to know
Coordinate geometry translates geometric relationships into algebra. Most questions reduce to a small toolkit: gradient, line equations, midpoint, distance and the relationship between parallel or perpendicular gradients.
- Calculate gradient from two points.
- Use point-gradient or y=mx+c form for straight lines.
- Use midpoint and distance formulas accurately.
- Recognise equal gradients for parallel lines and product -1 for perpendicular non-vertical lines.
Key mathematics
Gradient
m=\frac{y_2-y_1}{x_2-x_1}
Keep the point order consistent in numerator and denominator.Point-gradient form
y-y_1=m(x-x_1)
Useful when one point and the gradient are known.Distance
d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
Derived from Pythagoras.Midpoint
M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)
Average corresponding coordinates.Worked example
For $A(1,2)$ and $B(5,10)$, the gradient is $$m=\frac{10-2}{5-1}=2.$$ The midpoint is $(3,6)$ and the distance is $$\sqrt{4^2+8^2}=4\sqrt5.$$
EXAM TECHNIQUE
Exam tips
- Write the two points vertically before forming the gradient to avoid mixed ordering.
- Keep exact square-root distance answers unless a decimal is required.
- For perpendicular lines, find the negative reciprocal gradient carefully.
WATCH OUT
Common mistakes
- Swapping point order in only the numerator or denominator.
- Using $m_1+m_2=-1$ instead of $m_1m_2=-1$ for perpendicular lines.
- Averaging distances instead of averaging x- and y-coordinates separately for a midpoint.
Frequently asked questions
How do I find a line through two points?
First calculate the gradient, then use point-gradient form with either point.
What is the gradient of a perpendicular line?
For non-vertical lines it is the negative reciprocal, so the product of the gradients is minus one.
Should distance be exact?
Keep it exact in surd form unless the question asks for a decimal approximation.