Distance and section formulas on a grid
Place two points A and B on a coordinate grid, like two shops on a town map. Find the distance between them with Pythagoras, and the point P that divides AB in a ratio such as 1 : 2 or 1 : 1 (the midpoint).
Core concept
The distance AB = √((x₂ − x₁)² + (y₂ − y₁)²); the point dividing AB in the ratio m : n is ((m x₂ + n x₁)/(m + n), (m y₂ + n y₁)/(m + n)), and the midpoint is the case 1 : 1
Topic: Coordinate geometry
A point on a grid is written (x, y): x steps across, then y steps up. To find how far apart two points A(x₁, y₁) and B(x₂, y₂) are, draw the right-angled triangle under AB: its sides are x₂ − x₁ across and y₂ − y₁ up. By Pythagoras, AB = √((x₂ − x₁)² + (y₂ − y₁)²). A point P that divides AB in the ratio: A way to compare two amounts, like 2 cups of rice to 3 cups of water (2 : 3). All key words m : n is m parts of the way from A and n parts from B. Its coordinates: A pair of numbers (x, y) that tells you exactly where a point is on a grid. All key words are a weighted average: Add all the values and divide by how many there are. Also called the mean. All key words: P = ((m x₂ + n x₁)/(m + n), (m y₂ + n y₁)/(m + n)). When m = n = 1, P is the midpoint, halfway along: ((x₁ + x₂)/2, (y₁ + y₂)/2).
What you will learn
- Find the distance between two points with the distance formula.
- Find the point that divides a line segment in a given ratio.
- Find the midpoint of a line segment.
Before you start
- Plot points with negative and positive coordinates.
- Know Pythagoras' theorem.
Materials
- Graph paper, a ruler and a pencil
Equipment
- None
Procedure
- Record A(−2, 1), B(4, 4) and P dividing AB in 1 : 2.
- Change the ratio to 1 : 1 and record the midpoint.
- Move A to (0, 0) and B to (3, 4), and record.
- Change the ratio to 2 : 1 and record.
How this simulation is simplified
- One square of the grid is one unit; P lies on the straight line from A to B.