Coordinate Geometry

The Coordinate Plane

The coordinate plane (also called the Cartesian plane) is a flat surface defined by two perpendicular number lines:

• The x-axis runs horizontally (left-right)
• The y-axis runs vertically (up-down)
• They meet at the origin, coordinates (0, 0)

Every point on the plane is described by an ordered pair (x, y), where x tells you how far to move horizontally and y tells you how far to move vertically from the origin.

When plotting a point, always read x first, then y — a common memory aid is "along the corridor, then up the stairs."

The Four Quadrants

The axes divide the plane into four quadrants:

Quadrant	x	y	Example
I (top right)	positive	positive	(3, 5)
II (top left)	negative	positive	(-2, 4)
III (bottom left)	negative	negative	(-1, -3)
IV (bottom right)	positive	negative	(4, -2)

Points on the axes are not in any quadrant. The origin (0,0) is the intersection point.

Midpoint Formula

The midpoint of a line segment is the point exactly halfway between its two endpoints.

Formula: Midpoint = ((x₁ + x₂)/2 , (y₁ + y₂)/2)

Worked example: Find the midpoint of (2, 4) and (8, 10).

• x: (2 + 8)/2 = 10/2 = 5
• y: (4 + 10)/2 = 14/2 = 7
• Midpoint = (5, 7)

Think of it as finding the average of each coordinate separately.

Distance Formula

The distance between two points is found using Pythagoras' theorem applied to the horizontal and vertical differences.

Formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)

Worked example: Distance from (0, 0) to (3, 4).

• d = √((3-0)² + (4-0)²) = √(9 + 16) = √25 = 5

Worked example: Distance from (1,1) to (4,5).

• d = √((4-1)² + (5-1)²) = √(9 + 16) = √25 = 5

This is essentially creating a right-angled triangle between the two points and finding the hypotenuse.

Gradient Formula

The gradient (also called slope) measures how steep a line is. It is the ratio of the vertical change to the horizontal change between any two points on the line.

Formula: m = (y₂ - y₁)/(x₂ - x₁)

Worked example: Gradient of the line joining (1, 2) and (4, 11).

• m = (11 - 2)/(4 - 1) = 9/3 = 3

Gradient interpretation:

• Positive gradient — line slopes upward left to right
• Negative gradient — line slopes downward left to right
• Zero gradient — horizontal line
• Undefined gradient — vertical line

Equation of a Straight Line

The standard form is y = mx + c, where:

• m = gradient (slope of the line)
• c = y-intercept (where the line crosses the y-axis)

Worked example: A line has gradient 2 and passes through (0, -3).

• y = 2x + (-3) → y = 2x - 3

Finding the equation from two points:

1. Calculate the gradient using the gradient formula.
2. Substitute m and one point into y = mx + c to find c.

Example: Line through (1, 2) and (3, 8).

• Gradient: m = (8-2)/(3-1) = 6/2 = 3
• Using (1,2): 2 = 3(1) + c → c = -1
• Equation: y = 3x - 1

Parallel and Perpendicular Lines

Parallel lines have the same gradient.

• y = 4x + 1 and y = 4x - 5 are parallel (both have m = 4)

Perpendicular lines have gradients that are negative reciprocals of each other — they multiply to give -1.

• If line 1 has gradient 3, a perpendicular line has gradient -1/3
• Check: 3 × (-1/3) = -1 ✓

Key Terms

Term	Meaning
Origin	The point (0,0) where the axes meet
x-intercept	Where a line crosses the x-axis (y = 0)
y-intercept	Where a line crosses the y-axis (x = 0)
Gradient	The steepness of a line; rise over run
Midpoint	The point halfway between two coordinates
Distance	The length of the line segment between two points

Common Mistakes

• Reversing x and y — always plot (x, y): x is horizontal, y is vertical.
• Subtracting coordinates in different orders — in the gradient formula, use the same order for both numerator and denominator: (y₂ - y₁)/(x₂ - x₁), not mixed.
• Forgetting to square root in the distance formula — the formula gives d², not d, before you take the square root.
• Confusing parallel and perpendicular gradients — parallel lines have equal gradients; perpendicular lines have negative reciprocal gradients.
• Sign errors in the y-intercept — when c is negative, write y = mx - |c|, not y = mx + c with a positive number.

Tips and Tricks

• To check if a point lies on a line, substitute its x-value into the equation and verify the y-value matches.
• For the distance formula, draw a right-angled triangle sketch to visualise the horizontal and vertical legs before calculating.
• Memorise the gradient formula as "rise over run" — rise is the vertical change, run is the horizontal change.
• When finding parallel or perpendicular lines, write the gradient of the original line first, then derive the new one.