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.

Euclid built his entire geometric system around lines: the very first of his five postulates in the Elements (c. 300 BCE) states that a straight line can always be drawn between any two points, and the famous fifth postulate — about exactly one line through a point being parallel to a given line — puzzled mathematicians for over 2,000 years before it was finally shown, in the 1800s, that geometries without it are equally valid. The algebraic form y = mx + c that lets us write a line as an equation only became possible after coordinate geometry was introduced in the 17th century, fusing lines with algebra. Straight lines matter well beyond the page too: laser levels use a beam of light to mark a perfectly straight, horizontal line for builders, and railway engineers must calculate exact gradients so that tracks curve and rise smoothly enough for a train to travel safely.

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.

What is the gradient of the line y = 3x - 5?

The equation is in the form y = mx + c, so m = 3 and c = -5.

Are the lines y = 2x + 1 and y = 2x - 7 parallel?

Both have gradient m = 2. Same gradient, different intercepts, so yes, they are parallel.

Are y = 3x + 1 and y = -1/3 x + 4 perpendicular?

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: Gradient & Line Relationships

Find the Gradient