Coordinate geometry turns shapes into numbers, so gradients, distances and areas come from algebra. Questions combine several formulas, which makes choosing the right one the main skill.
How do the lessons connect?
Using gradient, midpoint and distance together is the base, because the other lessons reuse all three. Finding equations of parallel and perpendicular lines builds on the gradient.
Calculating areas from coordinates gives a method for any polygon. Solving coordinate locus problems then uses distances to describe a moving point. When a problem has more than one route, see geometry problems with several possible approaches.
How can one question have three routes?
Here is an original example. Show that the triangle with A(1, 2), B(4, 6) and C(8, 3) has a right angle at B.
- Gradients: AB = 4/3 and BC = −3/4, and 4/3 × (−3/4) = −1, so AB ⊥ BC.
- Distances: AB² = 25, BC² = 25 and AC² = 50, and 25 + 25 = 50, so Pythagoras holds.
- Vectors: AB = (3, 4) and BC = (4, −3), and the dot product is 12 − 12 = 0.
All three agree. Knowing several routes helps when one of them is awkward in a new question.
Where should you start?
- Formulas are new: begin with gradient, midpoint and distance, then the equation of a line.
- Lines are fine but areas are not: read the lesson on areas from coordinates.
- Close to an exam: try the coordinate geometry practice set and keep a mistake log.
When is tuition worth considering?
Free lessons show each method. A one-to-one teacher watches you meet a problem you have not seen and shows what in the wording points to a method.
Read online one-to-one SPM Additional Mathematics tuition for an example lesson, or return to the Additional Mathematics guide.