Coordinate geometry
Use coordinate differences consistently. Gradient measures vertical change divided by horizontal change. Distance uses the squares of both changes. A line equation must fit every point given on the line.
What this guide covers
Gradient, distance between points, straight-line equations and geometric problems using coordinates. Coordinate methods convert geometric information into calculations.
Syllabus reference: 2026 Mathematics 4052, G6, page 10. These are Mentora search groupings, not new official syllabus categories.
Worked example
Find the equation of the line through (2, 5) and (6, 13).
Gradient m = (13 − 5)/(6 − 2) = 2. In y = 2x + c, substitute (2, 5): 5 = 4 + c, so c = 1. The equation is y = 2x + 1; substituting x = 6 gives 13 as required.
Try three practice checks
Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.
Check 1
Find the gradient through (−1, 4) and (3, −4).
Show worked answer
m = (−4 − 4)/(3 − (−1)) = −8/4 = −2.
Check 2
Find the distance between (1, 2) and (4, 6).
Show worked answer
The changes are 3 and 4. Distance = √(3² + 4²) = 5 units.
Check 3
A line has gradient −3 and passes through (2, 1). Find its equation.
Show worked answer
Use y = −3x + c. Then 1 = −6 + c, giving c = 7 and y = −3x + 7.
Common mistakes to watch for
- Subtract coordinates in the same order in the numerator and denominator of a gradient.
- A vertical line has undefined gradient; division by zero is not zero gradient.
- The y-intercept c is not generally the y-coordinate of a supplied point unless x = 0.
If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.
You may need first
These verified prerequisite links suggest useful foundations. They do not make a claim about your current readiness.
- Pythagoras' theorem: finding the hypotenuseHorizontal and vertical changes form perpendicular sides; Pythagoras gives the distance between the points.Practise trigonometry →
- Gradient of a line from two coordinatesFind the gradient first, then substitute a given point into y = mx + c to obtain c.Practise coordinate geometry →
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More G3 Math topicsPublished by Mentora · Sources checked 11 Sept 2026
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