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Simply put, a line segment is just a piece of a given straight line. Each end of the line is called an endpoint. The two endpoints are usually denoted by letters, like A B or C D.
While AB is a common choice for endpoint labels, you could theoretically name them with any variable you like (e.g. FG or PQ). The usual notation is to write out the two endpoint labels with a line above them. For example: AB.
Directed Line Segment
A directed line segment has, like the name suggests, a direction. The straight line is drawn with an arrow pointing in a definite direction. Many quantities, like acceleration, force, or velocity, involve a magnitude and a direction, so directed line segments, like lines (2) and (3) in the above image, are used to represent them. To put this another way, directed line segments are vectors (Kishan, 2007).
The initial point is where the line begins. In the above image, the initial points are point B (image 2) and point A (image 3). The terminal point is where the line ends: point A (image 2) and point B (image 3). The initial and terminal points are not interchangeable, so and are not the same.
Directed line segments are equivalent if they have the same length and direction. You can show that two segments are equivalent with the distance formula (an application of the Pythagorean theorem):
and slope formula (rise over run).
Example question: Are these two line segments equivalent?
- A(0, 0) to B(3, 2)
- C(1, 2) to D(4, 4).
Step 1: Apply the distance formula to both line segments (Note: the double lines || indicate length):
. Both segments are the same length (radic;13).
Step 2: Use the slope formula for both segments:
Both segments have the same slope.
The two line segments have the same length and slope, so they are equivalent.
Endpoint on a Ray and Line Segment
A ray only extends indefinitely in just one direction. A ray is denoted with an arrow above the two endpoints. The initial point is where the line is limited (i.e. ends abruptly) and the terminal point is where the ray continues indefinitely. In ray notation, an arrow is placed above the two endpoints; the initial point comes first.
For example, the following image shows the ray :
Use of Endpoints
Endpoints are primarily used to find Riemann sums. The right-hand rule uses right endpoints for the calculation; The left-hand rule uses left endpoints.
The term is also used define points where a function simply ends.
An important note though, is that an endpoint in calculus isn’t usually a “point” in the usual sense of the word. It’s defined by a directional derivative or limit (i.e. values leading up to the endpoint, rather than the value at the endpoint itself).
Blank, B. & Krantz, S. (2006). Calculus: Multivariable, Volume 2. Key College Pub.
Kishan, H. (2007). Vector Algebra and Calculus. Atlantic Publishers & Distributors (P) Limited.
Stewart, J. (2015). Single Variable Calculus. Cengage Learning.
Introduction to Geometry: Rays and Angles
Stephanie Glen. "Line Segment: Definition, Endpoints, Examples, Equivalent" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/line-segment-equivalent/
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