The equation for the length of a line segment between two points ((x_1, y_1)) and ((x_2, y_2)) is:
[ L=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} ]
This is commonly called the distance formula. Think about it: it calculates the straight-line distance between two coordinates by applying the Pythagorean theorem. Here's the thing — a mathematical line extends infinitely in both directions, so it does not have a finite length. When discussing the “length of a line,” we usually mean the length of a line segment with two defined endpoints.
Introduction
Finding the length of a line segment is a fundamental skill in coordinate geometry. It is used in mathematics, physics, engineering, architecture, computer graphics, navigation, and many other fields.
The formula works because any diagonal line segment can be treated as the hypotenuse of a right triangle. The horizontal and vertical changes between its endpoints form the other two sides of that triangle Turns out it matters..
For two points:
[ A(x_1,y_1) \quad \text{and} \quad B(x_2,y_2) ]
the length of segment (AB) is:
[ \boxed{AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}} ]
The order of subtraction does not affect the answer because each difference is squared.
Understanding the Formula
The distance formula is:
[ L=\sqrt{(\Delta x)^2+(\Delta y)^2} ]
where:
- (L) is the length of the line segment.
- (\Delta x=x_2-x_1) is the horizontal change.
- (\Delta y=y_2-y_1) is the vertical change.
The Greek letter delta, (\Delta), means “change in.” Which means, (\Delta x) represents the change in the (x)-coordinates, while (\Delta y) represents the change in the (y)-coordinates.
This formula is directly connected to the Pythagorean theorem:
[ a^2+b^2=c^2 ]
If (a=\Delta x), (b=\Delta y), and (c=L), then:
[ (\Delta x)^2+(\Delta y)^2=L^2 ]
Taking the square root gives:
[ L=\sqrt{(\Delta x)^2+(\Delta y)^2} ]
Steps for Finding the Length of a Line Segment
To calculate the length of a line segment using two endpoints, follow these steps:
-
Identify the coordinates of both endpoints.
Label them as ((x_1,y_1)) and ((x_2,y_2)). -
Find the horizontal distance.
Calculate (x_2-x_1). -
Find the vertical distance.
Calculate (y_2-y_1). -
Square both differences.
Squaring removes negative signs and prepares the values for the Pythagorean relationship. -
Add the squared values.
-
Take the square root of the sum.
The result is the length of the line segment.
Here's one way to look at it: find the length of the segment connecting ((2,3)) and ((8,11)).
[ L=\sqrt{(8-2)^2+(11-3)^2} ]
[ L=\sqrt{6^2+8^2} ]
[ L=\sqrt{36+64} ]
[ L=\sqrt{100}=10 ]
The length of the line segment is 10 units.
Horizontal and Vertical Line Segments
The general distance formula works for every line segment, but horizontal and vertical segments can be calculated even more simply And that's really what it comes down to..
Horizontal Segments
A horizontal line segment has the same (y)-coordinate at both endpoints. For example:
[ (3,5) \quad \text{and} \quad (10,5) ]
Since there is no vertical change:
[ \Delta y=0 ]
The length is simply the absolute difference between the (x)-coordinates:
[ L=|x_2-x_1| ]
For the example:
[ L=|10-3|=7 ]
The segment is 7 units long That's the part that actually makes a difference..
Vertical Segments
A vertical line segment has the same (x)-coordinate at both endpoints. For example:
[ (4,2) \quad \text{and} \quad (4,9) ]
Since there is no horizontal change:
[ \Delta x=0 ]
The length is the absolute difference between the (y)-coordinates:
[ L=|y_2-y_1| ]
For the example:
[ L=|9-2|=7 ]
The segment is again **
