Coordinates turn geometry into algebra
The position of a point P in the Cartesian plane is written P(x, y). The x-coordinate gives its horizontal displacement from the y-axis, and the y-coordinate gives its vertical displacement from the x-axis.
Coordinate geometry lets us measure a figure without relying on an accurate drawing. The main tools are the distance formula, section formula and area formula.
Distance between two points
For A(x₁, y₁) and B(x₂, y₂), draw horizontal and vertical lines to form a right triangle. Its horizontal change is x₂ − x₁ and its vertical change is y₂ − y₁. The Pythagoras theorem gives:
AB = √((x₂ − x₁)² + (y₂ − y₁)²)
The order of subtraction does not matter because each difference is squared.
Example. The distance between A(2, 3) and B(5, 7) is
√((5 − 2)² + (7 − 3)²) = √(9 + 16) = 5 units.
Useful special cases:
distance of (x, y) from the origin = √(x² + y²),
points with the same x-coordinate are vertically aligned,
points with the same y-coordinate are horizontally aligned.
The section formula
Suppose P divides the segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio AP : PB = m : n. Then:
P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))
The weight beside each endpoint comes from the opposite part of the ratio. A point closer to B gives more weight to B's coordinates.
Example. A(1, 2) and B(7, 8) are divided in the ratio 2 : 1. Then
P = ((2 × 7 + 1 × 1)/3, (2 × 8 + 1 × 2)/3) = (5, 6).
The midpoint
The midpoint divides a segment in the ratio 1 : 1, so:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Area of a triangle
For A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃):
Area = ½ ∣x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)∣
The absolute value is necessary because area cannot be negative. Reversing the order of the vertices changes the sign inside but not the area.
Example. For A(0, 0), B(4, 0) and C(2, 3):
Area = ½ ∣0 + 4(3 − 0) + 2(0 − 0)∣ = 6 square units.
Testing collinearity
Three points are collinear when they lie on one straight line. Their triangle has no enclosed region, so the points are collinear exactly when the area formula gives 0.
Recognising common question types
Is a triangle isosceles? Find all three side lengths and compare them.
Is it a right triangle? Compare the squares of the side lengths using the Pythagoras theorem.
Find a point on an axis. Write it as (x, 0) on the x-axis or (0, y) on the y-axis, then use the given distance or ratio condition.
Find a missing coordinate. Substitute it into the distance, section or area formula and solve the resulting equation.
Keep squared distances until the final step when only a comparison is needed. This avoids unnecessary square roots.
Summary
Distance comes from the Pythagoras theorem.
The section formula gives a weighted average of endpoint coordinates.
The midpoint is the special case with ratio 1 : 1.
The coordinate area formula also tests collinearity: area zero means collinear points.