Line segment

In geometry, a line segment is a part of a straight line that has two distinct endpoints. These endpoints are the furthest points on the segment, and every point between them is also part of the segment. The length of a line segment is found by measuring the straight-line distance between its two endpoints, known as the Euclidean distance.

Line segments are important in many areas of geometry. For example, the sides of shapes like triangles and squares are line segments. When the endpoints of a line segment are corners (or vertices) of a shape, the segment can be an edge if the corners are next to each other, or a diagonal if they are not.

Line segments can also connect points on curves. If both endpoints lie on a curve, such as a circle, the line segment is called a chord. Understanding line segments helps us study and describe many geometric shapes and their properties.

In real or complex vector spaces

A line segment is a part of a straight line that has two endpoints. It includes every point between these two endpoints. In math, we can describe a line segment using special rules and symbols.

We can think of a line segment as a set of points that follow a certain pattern. This pattern helps us understand exactly which points are part of the line segment. There are also open line segments, which do not include the endpoints, and closed line segments, which do include them.

Properties

A line segment is a connected and non-empty part of a straight line, with two endpoints that mark where it starts and ends. In certain types of spaces, a closed line segment is also a closed set, while an open line segment can be an open set if the space is one-dimensional. Line segments can have different relationships with each other, such as intersecting, being parallel, or being skew. Unlike full lines, two line segments in the same plane do not always have to cross each other if they are not parallel.

Main article: Line segment

In proofs

Segments are important in geometry and help us understand shapes and proofs. In convex sets, any line segment connecting two points stays inside the set. This idea helps change some difficult problems about shapes into simpler ones about line segments.

The segment addition postulate lets us add together lengths of segments that are the same. This helps us prove that some segments are the same length as others.

As a degenerate ellipse

A line segment can be thought of as a special kind of ellipse. In this special case, the smaller half of the ellipse gets smaller and smaller until it disappears. The two special points of the ellipse, called foci, end up at the ends of the line segment. When you travel around this special ellipse, you go along the line segment two times. This special path is called a radial elliptic trajectory.

In other geometric shapes

Line segments are found in many different geometric shapes. In triangles, important segments include the altitudes, medians, perpendicular bisectors, and angle bisectors. These segments connect different points such as vertices and midpoints.

In quadrilaterals, line segments appear as sides, diagonals, and special segments like bimedians and maltitudes. In circles and ellipses, a line segment connecting two points on the shape is called a chord. Special chords in circles include the diameter and radius. In ellipses, there are major and minor axes, as well as semi-major and semi-minor axes.

Directed line segment

Further information: Orientation (vector space) § On a line

See also: Relative position

When a line segment is given an orientation (direction), it is called a directed line segment or oriented line segment. This shows a possible translation or displacement, like something might move because of a force. In math and physics, this idea helps us understand how things can change in both size and direction.

Generalizations

Just like straight line segments, we can also talk about pieces of curves, called arcs. In a simple one-dimensional space, a special shape called a ball is actually just a line segment. There are also ideas like oriented plane segments or bivectors that build on the idea of directed line segments. In more complex geometries, geodesic segments act like line segments. A line segment is a type of shape called a one-dimensional simplex; the next step up, a two-dimensional simplex, is a triangle.

Types of line segments

A line segment can have special names depending on where it is found. For example, a chord is a line segment that connects two points on a circle, but does not go through the center of the circle. The diameter is a special chord that passes through the center of the circle, making it the longest chord possible. The radius is a line segment from the center of the circle to any point on its edge. These are just a few types of line segments that appear in geometry.

Main articles: Chord (geometry), Diameter, Radius