Coordinate Geometry

Coordinates and gradient

Every point is fixed by two numbers, and every straight line reduces to two more: how steeply it climbs, and where it crosses the axes.

Coordinates and straight-line graphs

A point is written as an ordered pair (x, y). A straight-line graph is usually given as an equation — most often y = mx + c, where m is the gradient and c is the y-intercept.

Finding the gradient

The gradient of a line joining two points is the change in y divided by the change in x: m = (y₂ − y₁) / (x₂ − x₁). Mixing up the order of subtraction is the most common mistake.

Worked example: a line passes through (1, 3) and (5, 11). m = (11 − 3) ÷ (5 − 1) = 8 ÷ 4 = 2 — the line climbs 2 units for every 1 unit moved right.

Length and midpoint

Length and midpoint both start from the same two coordinates — one measures the distance between them, the other finds the point exactly between them.

Length of a line segment

The segment is the hypotenuse of a right triangle formed by the horizontal and vertical differences between its endpoints: d = √[(x₂ − x₁)² + (y₂ − y₁)²].

Midpoint of a line segment

The midpoint is the average of the two x-coordinates, and the average of the two y-coordinates: ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ).

Worked example: A(−2, 3) and B(4, −5) are the endpoints of a segment. AB = √[(4 − (−2))² + (−5 − 3)²] = √(6² + (−8)²) = √(36 + 64) = √100 = 10. Midpoint = ( (−2 + 4)/2 , (3 + (−5))/2 ) = (1, −1). Bracketing negative coordinates before subtracting keeps both calculations sign-safe.

Equations of a line

Once two points — or a gradient and a point — are known, the equation of the line through them is fully determined. Cambridge always wants that equation fully simplified, in the form the question specifies.

Finding an equation from two points

Find the gradient first, then substitute one point into y = mx + c to find c. Worked example: a line through (−1, −2) and (3, 6). m = (6 − (−2)) ÷ (3 − (−1)) = 8 ÷ 4 = 2. Substituting (3, 6) into y = 2x + c gives c = 0, so y = 2x. Writing c = 0 explicitly avoids the common slip of dropping the constant term altogether.

ExtendedReading a gradient and intercept from ax + by = c

This form hides m and c until it is rearranged. Worked example: find the gradient and y-intercept of 5x + 4y = 8. Rearranging: 4y = −5x + 8, so y = −5/4 x + 2. The rearranged form now matches y = mx + c directly: m = −5/4, c = 2.

Parallel and perpendicular lines

Equal gradients mean parallel; gradients multiplying to −1 mean perpendicular: m₁ = m₂ for parallel lines, and m₁ × m₂ = −1 for perpendicular lines.

Parallel lines

A parallel line has the same gradient — only the y-intercept differs. Worked example: the line parallel to y = 3x − 2 through (2, 5) has m = 3; substituting gives 5 = 3(2) + c, so c = −1, and the line is y = 3x − 1.

Perpendicular lines and the perpendicular bisector

A perpendicular gradient is the negative reciprocal of the original. Worked example: the perpendicular bisector of the segment joining (−3, 8) and (9, −2). Midpoint M = (3, 3). Gradient of the segment m = (−2 − 8) ÷ (9 − (−3)) = −10 ÷ 12 = −5/6, so the perpendicular gradient is 6/5. Substituting M into y = (6/5)x + c: 3 = (6/5)(3) + c, so c = −3/5, giving y = (6/5)x − 3/5.