What you'll learn
Coordinate geometry connects algebra and geometry by placing points and lines on the Cartesian plane, and it is a core part of the CSEC Mathematics syllabus under Relations, Functions and Graphs. Using coordinates, you can calculate the length of a line segment, find its midpoint, measure its gradient (slope), and write the equation of a straight line. In this guide you will learn each of these tools, how they fit together, and how to handle parallel and perpendicular lines. These skills underpin graph work, linear functions and later study, and they appear in both Paper 1 and Paper 2, often combined in a single multi-part question.
Key terms and definitions
Cartesian plane — the grid formed by the horizontal x-axis and vertical y-axis, meeting at the origin (0, 0).
Coordinates — an ordered pair (x, y) giving a point's position.
Gradient (slope) — the steepness of a line, measured as the change in y divided by the change in x.
Midpoint — the point exactly halfway between two given points.
Length (distance) — the straight-line distance between two points.
Intercept — where a line crosses an axis; the y-intercept is the value of y when x = 0.
y = mx + c — the equation of a straight line, with gradient m and y-intercept c.
Core concepts
The distance (length) of a line segment
The length between points (x₁, y₁) and (x₂, y₂) comes from Pythagoras' theorem applied to the horizontal and vertical gaps:
length = √[(x₂ − x₁)² + (y₂ − y₁)²].
The horizontal change is (x₂ − x₁) and the vertical change is (y₂ − y₁); these are the legs of a right-angled triangle whose hypotenuse is the segment itself.
The midpoint
The midpoint is the average of the coordinates:
midpoint = ( (x₁ + x₂) ÷ 2 , (y₁ + y₂) ÷ 2 ).
Simply average the x-values and average the y-values. This is widely used to find centres of lines and shapes.
The gradient
The gradient measures steepness:
gradient m = (y₂ − y₁) ÷ (x₂ − x₁) = "rise over run".
A positive gradient rises from left to right; a negative gradient falls. A horizontal line has gradient 0, and a vertical line has an undefined gradient. Be careful to subtract the coordinates in the same order on top and bottom.
The equation of a straight line
Every straight line can be written as y = mx + c, where m is the gradient and c is the y-intercept. To find the equation, you need the gradient and one point. Substitute the gradient for m and the point's coordinates for x and y to solve for c; or use y − y₁ = m(x − x₁) directly. Then write the finished equation.
Parallel and perpendicular lines
Two lines are parallel when they have the same gradient (m₁ = m₂). Two lines are perpendicular when the product of their gradients is −1 (m₁ × m₂ = −1); equivalently, one gradient is the negative reciprocal of the other. For example, a line perpendicular to one with gradient 2 has gradient −½.
Worked examples
Example 1: Length and midpoint (Paper 2 style)
Find the length and midpoint of the segment joining A(1, 2) and B(7, 10).
Length = √[(7 − 1)² + (10 − 2)²] = √[6² + 8²] = √[36 + 64] = √100 = 10 units. Midpoint = ( (1 + 7)÷2 , (2 + 10)÷2 ) = (4, 6). So the midpoint is (4, 6).
Example 2: Equation of a line (Paper 2 style)
Find the equation of the line through P(2, 3) with gradient 4.
Using y = mx + c with m = 4: 3 = 4(2) + c, so 3 = 8 + c, giving c = −5. The equation is y = 4x − 5.
Example 3: Perpendicular line (Paper 2 style)
A line L has equation y = 2x + 1. Find the equation of the line perpendicular to L passing through (4, 3).
The gradient of L is 2, so the perpendicular gradient is the negative reciprocal, −½. Using y = mx + c: 3 = −½(4) + c = −2 + c, so c = 5. The equation is y = −½x + 5.
Common mistakes and how to avoid them
Subtracting coordinates inconsistently. For gradient, if you take y₂ − y₁ on top you must take x₂ − x₁ (same order) on the bottom.
Confusing midpoint with gradient. Midpoint adds and halves; gradient subtracts and divides. Keep the two formulas distinct.
Forgetting to square inside the distance formula. The differences must be squared before adding, and you take the square root at the end.
Getting the perpendicular gradient wrong. It is the negative reciprocal: flip the fraction and change the sign (2 → −½, −¾ → 4/3).
Not solving for c. The equation is incomplete until you substitute a point to find the y-intercept.
Exam technique for Coordinate Geometry
Label your points. Write (x₁, y₁) and (x₂, y₂) above the coordinates so you substitute consistently.
Quote the formula first. Stating "gradient = (y₂ − y₁) ÷ (x₂ − x₁)" earns method marks even if arithmetic slips.
Use a quick sketch. Plotting the points roughly helps you check whether a gradient should be positive or negative and whether an answer is reasonable.
Remember the gradient rules. Parallel → equal gradients; perpendicular → product of gradients is −1.
Finish the equation. Always present the final line in the form y = mx + c with both m and c found.
Quick revision summary
Coordinate geometry works with points (x, y) on the Cartesian plane. The length of a segment is √[(x₂ − x₁)² + (y₂ − y₁)²], from Pythagoras. The midpoint is the average of the coordinates, ( (x₁ + x₂)÷2 , (y₁ + y₂)÷2 ). The gradient is (y₂ − y₁) ÷ (x₂ − x₁), "rise over run", positive for an uphill line and negative for a downhill one. The equation of a straight line is y = mx + c, where m is the gradient and c the y-intercept; find c by substituting a known point. Parallel lines share the same gradient, while perpendicular lines have gradients whose product is −1 (negative reciprocals). Label your points, quote each formula before substituting, keep the midpoint (add-and-halve) separate from the gradient (subtract-and-divide), and always present the final straight-line equation fully simplified.