Coordinate Geometry

Points on a plane: the distance between two points, and the point that divides a line segment in a given ratio.

Key ideas

• The distance formula comes from Pythagoras: the horizontal and vertical gaps are the two legs.
• A point dividing AB in the ratio m : n (inside AB) is found with the section formula.
• The midpoint is the section formula with m = n = 1.
• Three points A, B, C lie on one line when the two shorter distances add up to the longest.
• To show a quadrilateral is a square, rhombus or rectangle, compare its sides and diagonals using the distance formula.

Remember

Watch out

• Swapping m and n in the section formula: m goes with the second point (x₂, y₂).
• Forgetting that the square of a negative difference is positive.
• Stopping at (x₂ − x₁)² + (y₂ − y₁)² and forgetting the square root.

Worked example