Lines – Paths That Define Shape and Direction
A line is one of the most basic ideas in geometry. Lines form the boundaries of shapes, define directions, and underpin almost every geometric concept you will encounter.
Types of Straight Paths
| Type | Description | Has Two Endpoints? | Extends Forever? |
|---|---|---|---|
| Line | Straight path, no endpoints | No | Both directions |
| Line Segment | Part of a line with two endpoints | Yes | No |
| Ray | Starts at a point, extends one way | One endpoint | One direction |
Notation
A line through points A and B is written as line AB (with a double arrow above). A line segment from A to B is written as AB (with a bar above). A ray starting at A through B is written as AB (with a single arrow above).
Relationships Between Lines
| Relationship | Meaning | Key Property |
|---|---|---|
| Parallel lines | Never meet, always same distance apart | Same gradient (slope) |
| Perpendicular lines | Meet at exactly 90 degrees | Gradients multiply to give -1 |
| Intersecting lines | Cross at one point | Share exactly one point |
| Coincident lines | Lie exactly on top of each other | Identical equations |
Equations of Lines
On a coordinate grid, every straight line can be described by an equation. The most common form is y = mx + c, where m is the gradient and c is the y-intercept.
The equation is in the form y = mx + c, so m = 3 and c = -5.
Both have gradient m = 2. Same gradient, different intercepts, so yes, they are parallel.
Product of gradients: 3 times (-1/3) = -1. Product equals -1, so yes, they are perpendicular.
Length of a Line Segment
Use the distance formula. For segment AB where A = (x₁, y₁) and B = (x₂, y₂):
Length = √[(x₂ − x₁)² + (y₂ − y₁)²]
Key Takeaways
- A line extends forever; a line segment has two endpoints; a ray has one endpoint.
- Parallel lines have the same gradient and never meet.
- Perpendicular lines meet at 90 degrees and their gradients multiply to -1.
- The equation y = mx + c describes any straight line on a coordinate grid.
Practice Questions
- Name three real-life examples of line segments.
- What is the gradient of the line connecting (2, 3) and (6, 11)?
- Write the equation of a line parallel to y = 4x - 2 that passes through (0, 5).
- Write the equation of a line perpendicular to y = 2x + 1 that passes through the origin.
- Find the length of the segment joining P(1, 4) and Q(7, 12).