Co-ordinate Geometry — CIE IGCSE Mathematics Revision Notes
What you'll learn
- How to find the gradient, length and midpoint of a line segment from two points
- The equation of a straight line in the form y = mx + c, and how to build it from the information given
- How parallel and perpendicular lines relate through their gradients
- How to find the equation of a line through a point, parallel or perpendicular to another
- How to read and interpret gradients and intercepts in context
- The standard errors in this topic and how to avoid every one of them
Key terms and definitions
Co-ordinates — a pair (x, y) fixing a point, with x measured horizontally and y vertically, always in that order.
Gradient — the steepness of a line, measured as the change in y divided by the change in x.
y-intercept — where a line crosses the y-axis, the value of y when x = 0.
x-intercept — where a line crosses the x-axis, the value of x when y = 0.
Midpoint — the point exactly halfway along a line segment.
Line segment — the finite piece of a line between two named points.
Parallel — lines with equal gradients, which never meet.
Perpendicular — lines meeting at 90°, whose gradients multiply to −1.
Negative reciprocal — the result of turning a fraction upside down and changing its sign. The gradient of a perpendicular line.
Collinear — three or more points lying on the same straight line.
Core concepts
The three formulae
Every question in this topic is built from three results, and they all come from the same two points, A(x₁, y₁) and B(x₂, y₂).
Gradient: m = (y₂ − y₁) ÷ (x₂ − x₁)
Length: AB = √((x₂ − x₁)² + (y₂ − y₁)²)
Midpoint: ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
The length formula is Pythagoras' theorem applied to the horizontal and vertical gaps between the points, which is worth remembering because it means you can always rebuild it if the formula escapes you. The midpoint is simply the average of the x values and the average of the y values.
Two points about signs. Subtract in a consistent order — if you start with B's y value you must also start with B's x value, or the gradient comes out with the wrong sign. And in the length formula the squaring removes any sign problem, so a length is always positive.
What a gradient tells you
A positive gradient rises left to right; a negative gradient falls. A gradient of zero is a horizontal line, y = c. A vertical line has an undefined gradient and its equation is x = c — it cannot be written as y = mx + c at all, which is a distinction examiners test.
The size of the number is the steepness: a gradient of 4 is four times as steep as a gradient of 1.
The equation of a straight line
The standard form is y = mx + c, where m is the gradient and c is the y-intercept.
Questions give you the information in one of three ways, and each has a direct route:
Given the gradient and the y-intercept — substitute straight into y = mx + c.
Given the gradient and any point — substitute the gradient and the point's coordinates into y = mx + c and solve for c.
Given two points — find the gradient first, then use either point to find c.
An equation may also be presented rearranged, such as 2y + 6x = 5. Make y the subject before reading anything off it: y = −3x + 2.5, so the gradient is −3, not 6 or 2.
Parallel and perpendicular lines
Parallel lines have equal gradients. A line parallel to y = 4x − 7 has gradient 4, whatever its intercept.
Perpendicular gradients multiply to −1, which means each is the negative reciprocal of the other. If m₁ = 3 then m₂ = −1/3. If m₁ = −2/5 then m₂ = 5/2.
Two checks that catch most errors: the two gradients must have opposite signs, and if one is steep the other must be shallow. A gradient of 3 paired with −3 is not perpendicular, because 3 × −3 = −9.
Putting it together
A typical extended question asks for the equation of a line through a given point, perpendicular to a given line. The sequence never varies:
- Read the gradient of the given line, rearranging to y = mx + c first if necessary.
- Take the negative reciprocal for the perpendicular gradient.
- Substitute that gradient and the given point into y = mx + c.
- Solve for c and write the full equation.
Where two lines cross
The point where two lines intersect satisfies both equations at once, so finding it is solving them simultaneously.
With both lines in the form y = mx + c, set the two expressions equal to each other, solve for x, then substitute back into either equation to get y. Substituting into the other equation afterwards is a free check: both must give the same y.
Two lines fail to cross only if they are parallel. If you set the equations equal and the x terms cancel to leave a false statement, that is what has happened — the lines have the same gradient and never meet.
Proving geometric properties with coordinates
A common extended question gives three or four points and asks you to prove something about the shape they form. Each property has a specific test, and knowing which one to reach for is most of the work.
Collinear points — three points lie on the same line if the gradient between the first and second equals the gradient between the second and third.
A right angle — show the two gradients multiply to −1.
A parallelogram — show both pairs of opposite sides have equal gradients.
A rhombus — a parallelogram whose four sides are equal in length.
An isosceles triangle — show two side lengths are equal.
The diagonals of a parallelogram bisect each other — show the midpoint of one diagonal equals the midpoint of the other.
State the conclusion in words once the calculation is done. A page of correct arithmetic that never says "therefore the lines are perpendicular, so angle ABC is a right angle" leaves the final mark unclaimed.
Gradients in context
Coordinate geometry appears in applied questions where the axes carry units, and there the gradient is a rate: cost per item, distance per second, litres per minute. The y-intercept is the value when the other quantity is zero — a fixed charge, a starting distance, an initial volume.
Reading these correctly means always checking the axis labels before interpreting a number.
Worked examples
Example 1: Gradient, length and midpoint
A is (1, 2) and B is (5, 10). Find the gradient of AB, its length, and its midpoint.
Gradient: m = (10 − 2) ÷ (5 − 1) = 8 ÷ 4 = 2
Length: the horizontal gap is 5 − 1 = 4 and the vertical gap is 10 − 2 = 8.
AB = √(4² + 8²) = √(16 + 64) = √80 = 4√5 ≈ 8.94 (3 s.f.)
Midpoint: ((1 + 5) ÷ 2, (2 + 10) ÷ 2) = (3, 6)
A quick sanity check: the midpoint should look halfway between the two points, and (3, 6) sits between (1, 2) and (5, 10) on both coordinates.
Example 2: The equation of a line through a point
Find the equation of the line with gradient −½ passing through (2, 3).
Substitute m = −½ and the point into y = mx + c:
3 = −½(2) + c 3 = −1 + c c = 4
The equation is y = −½x + 4.
Check it: at x = 2, y = −½(2) + 4 = −1 + 4 = 3 ✓
Example 3: A perpendicular line
Find the equation of the line perpendicular to y = 3x + 1 that passes through (6, 2).
The given line has gradient 3, so the perpendicular gradient is the negative reciprocal, −1/3.
Substitute that gradient and the point (6, 2):
2 = −⅓(6) + c 2 = −2 + c c = 4
The equation is y = −⅓x + 4.
Two checks. The gradients multiply to 3 × −⅓ = −1 ✓. And substituting x = 6 gives −2 + 4 = 2, the correct y value ✓
Example 4: Finding an intersection
Where do y = 2x − 1 and y = −x + 5 cross?
Set them equal: 2x − 1 = −x + 5
3x = 6, so x = 2
Substitute into the first equation: y = 2(2) − 1 = 3
The lines cross at (2, 3). Check in the second equation: −2 + 5 = 3 ✓
Example 5: Proving a right angle
A is (0, 1), B is (3, 7) and C is (9, 4). Show that angle ABC is a right angle.
Gradient of AB = (7 − 1) ÷ (3 − 0) = 6 ÷ 3 = 2
Gradient of BC = (4 − 7) ÷ (9 − 3) = −3 ÷ 6 = −½
Multiplying: 2 × −½ = −1
Since the gradients of AB and BC multiply to −1, the two lines are perpendicular, and therefore angle ABC is a right angle.
Note that the conclusion is stated explicitly. The arithmetic alone does not answer a question that says "show that".
Common mistakes and how to avoid them
Subtracting inconsistently. Using y₂ − y₁ over x₁ − x₂ gives the gradient the wrong sign. Fix the order of your points before you start and keep it.
Reversing the coordinates. Always x first, then y. On a graph, across before up.
Reading the gradient off an unrearranged equation. In 2y = 8x + 6 the gradient is 4, not 8. Divide through first.
Getting the perpendicular gradient wrong. It is the negative reciprocal, not just the negative. For m = 3 the answer is −1/3, never −3.
Giving a negative length. Distance is always positive; the squaring in the formula guarantees it.
Forgetting that vertical lines are different. A vertical line has equation x = c and no gradient. It cannot be written in the form y = mx + c.
Rounding partway through. Keep the surd or the full decimal in your calculator until the final answer, then round once.
Exam technique for Co-ordinate Geometry
This topic rewards neatness more than most, because almost every error in it is a sign or a substitution slip rather than a misunderstanding.
Write the formula down before substituting. It earns the method mark even if the arithmetic then goes wrong, and it makes a sign error visible when you check.
Sketch it. A rough diagram with the two points marked takes fifteen seconds and immediately shows whether a gradient should be positive or negative. Answers that contradict the sketch are wrong.
State the gradient explicitly when a question asks for an equation. Examiners award a mark for the correct gradient separately from the full equation.
Leave surds exact unless told otherwise. √80 simplifies to 4√5, and an exact answer is safer than a rounded one unless the question specifies decimal places.
Check perpendicular answers by multiplying. If the two gradients do not give −1, something has gone wrong and you have ten seconds to find it.
Use the point you were given. Substituting it back into your final equation verifies the whole answer in one line.
Quick revision summary
- Gradient m = (y₂ − y₁) ÷ (x₂ − x₁) — subtract in a consistent order
- Length = √((x₂ − x₁)² + (y₂ − y₁)²), which is just Pythagoras, and always positive
- Midpoint = the average of the x values and the average of the y values
- Straight line: y = mx + c, with m the gradient and c the y-intercept
- Rearrange to y = mx + c before reading a gradient off an equation
- Parallel lines have equal gradients
- Perpendicular gradients multiply to −1 — take the negative reciprocal, not just the negative
- Horizontal line y = c; vertical line x = c with no gradient at all
- In applied questions the gradient is a rate and the intercept is the starting value
- Two lines cross where their equations are equal — solve simultaneously, then check in the other equation
- To prove a shape: equal gradients for parallel, product −1 for perpendicular, equal lengths for isosceles, equal midpoints for bisecting diagonals — and state the conclusion in words
- Sketch the points, write the formula, then substitute — most lost marks here are sign slips