Line segment

In geometry, a line segment is a part of a straight line with two special points called endpoints. These endpoints are the farthest points on the segment. Every point between them is also part of the segment. We can find the length of a line segment by measuring the straight-line distance between its two endpoints. This distance is called the Euclidean distance.

Line segments are useful in many parts of geometry. For example, the sides of shapes like triangles and squares are line segments. When the endpoints of a line segment are corners 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 are on a curve, such as a circle, the line segment is called a chord. Learning about line segments helps us understand and describe many geometric shapes and their properties.

In real or complex vector spaces

A line segment is a part of a straight line with 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 group of points that follow a pattern. This pattern tells us 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 piece of a straight line that has two endpoints. These points mark where the line segment starts and ends. In some spaces, a closed line segment is also a closed set. Line segments can have different relationships, like intersecting or being parallel. Unlike full lines, two line segments in the same plane do not always cross each other if they are not parallel.

Main article: Line segment

In proofs

Segments are important in geometry. They help us understand shapes and solve problems.

If you connect two points with a line segment inside a convex set, the whole segment stays inside that set. This makes some hard shape problems easier to solve by using line segments.

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

As a degenerate ellipse

A line segment is a special type of ellipse. In this 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 move along the line segment two times. This special path is called a radial elliptic trajectory.

In other geometric shapes

Line segments are parts of many geometric shapes. In triangles, important segments include altitudes, medians, perpendicular bisectors, and angle bisectors. These segments connect different points such as corners and midpoints.

In quadrilaterals, line segments appear as sides, diagonals, and special segments. 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 has an orientation (direction), it is called a directed line segment or oriented line segment. This shows how something might move because of a force. In math and physics, this idea helps us understand how things can change.

Generalizations

Just like straight line segments, we can also talk about pieces of curves, called arcs. In a simple space, a special shape called a ball is really 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 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