Contents (Click to skip to that section):
- Line Segment Definition
- Directed Line Segment Definition
- Equivalent Directed Line Segments
- Problem Solving: Length of a Line Segment
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.
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).
The length of a line segment between two points is calculated with the distance formula:
This formula only works for linear (straight line) functions. If you have a curved function, use the arc length formula instead.
Length of a Line Segment: Example
Example question: What is the length of the line segment for f(x) = 5x + 2 on the interval [0,1]?
Step 1: (Optional) Draw a Graph of the function. This step can help you see where the points you need for the formula are (x1, x2, y1, y2). I used Desmos.com to create this graph:
A quick look at the line segment between these two points and I’m going to guess that the length should be around 5. That gives me a way to check that my final answer is reasonable.
Step 2: Plug your two (x, y) coordinates into the distance formula.
If you’re having trouble deciding which point goes where, the following graph has the x and y points labeled.
Step 3: Simplify and solve:
- (1 – 0)2 = 1
- (7 – 2)2 = 25
- 1 + 25 = 26
- √(26) ≈ 5.1
So my guess at 5 was fairly close.
Length of a Line Segment & Pythagorean Theorem
You might notice that the distance formula is really an application of the Pythagorean theorem. And in fact, you could get to the same answer by using the Pythagorean formula a2 + b2 = c2.
The following image shows that the hypotenuse of the triangle lies on the line segment you’re trying to find the length of:
You may be wondering why we use the distance formula to find the length of a line segment instead of the more simple Pythagorean formula. The answer is that while a, b, and c work perfectly well in geometry, it doesn’t translate well to the Euclidean plane of x’s and y’s, especially once you get to variations on the distance formula for curves. You can’t fit a triangle to a curve, but you can fit a series of small lines. This process of fitting small lines to approximate a curve uses integration, and (along with derivatives) it’s one of the two fundamental areas of calculus.
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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