Points – The Starting Block of All Geometry
A point is the most fundamental object in all of geometry. Everything else — lines, angles, shapes — is built from points. Understanding what a point is, and how to work with points, is the first step in any geometry course.
What Is a Point?
A point is an exact location in space. It has no size, no width, no length, and no thickness. It is simply a position. In diagrams, a point is shown as a small dot and labelled with a capital letter such as A, B, or P.
Notation
| Symbol | Meaning | Example |
|---|---|---|
| A | A single point labelled A | Point A is at position (3, 2) |
| (x, y) | Coordinates on a grid | (5, -1) means 5 right, 1 down |
| (x, y, z) | Coordinates in 3D space | (2, 4, 1) in three dimensions |
Points on a Coordinate Grid
When we place geometry on a number grid (the coordinate plane), every point can be described by a pair of numbers (x, y). The x-value tells you how far to go right or left; the y-value tells you how far to go up or down.
Start at the origin (0, 0). Move 4 units to the right, then 3 units up. Mark the location and label it P.
Start at the origin. Move 2 units to the left, then 5 units up. Mark and label Q.
Collinear Points
Three or more points are collinear if they all lie on the same straight line. If they do not, they are non-collinear.
Check if the gradient between each pair is the same. A to B: (4-2)/(3-1) = 1. B to C: (6-4)/(5-3) = 1. Same gradient, so yes, they are collinear.
Distance Between Two Points
The distance between point A(x₁, y₁) and point B(x₂, y₂) is found using the distance formula, which comes directly from Pythagoras' theorem:
Distance = square root of [(x₂ − x₁)² + (y₂ − y₁)²]
Distance = √[(4−1)² + (5−1)²] = √[9 + 16] = √25 = 5 units.
Key Takeaways
- A point is an exact position in space with no size or dimension.
- Points are labelled with capital letters and located by coordinates (x, y).
- Collinear points all lie on the same straight line.
- The distance between two points uses the distance formula based on Pythagoras.
Practice Questions
- Plot the points A(2, 5), B(-3, 1), and C(0, -4) on a coordinate grid.
- Are points P(0, 0), Q(2, 4), and R(3, 5) collinear? Show your working.
- Find the distance between M(3, 7) and N(7, 4).
- A point lies on the y-axis and is 6 units from the origin. What are its possible coordinates?
- Point X is at (1, 2) and point Y is at (a, 8). The distance XY is 10 units. Find the value of a.