Mentora
E-Math resources

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.

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.

Useful foundations to check

These connections suggest what may help with this skill. Selecting a topic says nothing about your current attainment.

Applications and next topics

Use equal gradients to check parallel lines and distances to compare side lengths. Vectors describe the same coordinate changes as directed movements.

Choose a useful next resource

To understand a recurring error rather than relearn the whole skill: Check coordinate and gradient errors.

Explore the foundations connected to coordinate geometry

Sources and review

Original Mentora public practice, authored for this guide. Worked answers and syllabus mapping checked during editorial QA on 2026-09-11; no independent expert review is claimed. Graph connections are Mentora learning hints, not an official SEAB sequence. Read the source and examination-year notes.