A Line is a one-dimensional geometric figure that exists only in length without width and thickness. A Line comprises a group of points; it extends infinitely in opposite directions or combines with other lines to form various shapes. In a two-dimensional plane, a line is uniquely defined by a pair of points. Points situated on the same line are termed collinear.
In this article, we will discuss the introduction, definition of Line, and its meaning. We will also understand the different types of Lines and various equations related to Lines. We will also solve various examples and provide practice questions for a better understanding of the concept of this article.
Line in Geometry
Lines serve as the foundational elements of geometry. It plays an important role in understanding mathematical concepts. A line represents an infinitely extended, straight, one-dimensional pathway without endpoints in both directions. Typically, It is represented by a lowercase letter (e.g., ‘l’) or defined by two points (e.g., ‘AB’). Daily life examples of a line can be seen in railway tracks and straight roads.
The credit for introducing the concept of a line goes to Euclid, the esteemed Greek mathematician fondly known as the Father of Geometry. Euclid characterized a line as “a breadthless length”.
Definition of a Line
A Line is created by a collection of points moving along a straight trajectory but in opposing directions. Without breadth, depth, or curvature, a line keeps moving in a straight path endlessly in opposite directions. An absence of termination characterizes this continuous motion.
Line Segment
A Line Segment is defined as a line with two fixed endpoints. A line segment maintains a constant length, representing the distance between its two endpoints.
Ray
A Ray is defined as a line which has fixed starting point but lacks a defined endpoint. A ray can extend infinitely in a singular direction. Due to the absence of an endpoint, measurement the length of a ray is not applicable. For examples, a sunray or the light emitted from a torch, where the sun or the torch serves as the originating point, and the ray extends indefinitely.
Difference Between Line, Line Segment and Ray
Line, Line Segment and Rays are fundamental for geometrical figure. They sound and look similar but have significant difference between them. Let’s learn the difference between Line, Line segment and Rays in tabular form:
|
Line |
Line Segment |
Ray |
|---|---|---|
|
A Line is a collection of points that go forever in two opposite directions indefinitely. It is an endless continuous and it has no finite endpoints |
Line Segment is a part of the line that can link two fixed or definite endpoints, in which all points exist between these end points. |
When a line has a starting point but not a finite end point means end at infinity is called a Ray. |
|
It is represented with arrows at both ends to show that it extends infinitely in both directions. |
It has a definite length and it does not extend infinitely in both the direction. |
It represents one start point and an arrow at the other end that means it moves forever in one direction. |
|
It is represented by ↔. |
It is represented by a bar ‘―’ on the top of two endpoints. |
It is represented by →. |
Types of Lines
In geometry, various types of Lines exist, each with distinct characteristics. Let’s understand these Lines:
- Straight Line: A Straight Line is a group of collinear points without any curves, extending infinitely in opposite directions.
- Curved Line: Curved Line is a group of collinear points with curvature or bends in its configuration. For example, curve line can be seen in the meandering course of a river.
- Oblique or Slanting Line: Oblique or Slanting Lines are lines drawn in a slanting position, forming angles other than 0, 90, 180, 270, or 360 degrees with horizontal or vertical lines.
- Horizontal Line: Horizontal Line is a straight line extending from left to right or vice versa. It is also known as a sleeping line. Horizontal line can be seen lines on notebook paper, and table edges.
- Vertical Line: Vertical Line extends from top to bottom or bottom to top, known as a standing line. Anything forming a right angle with the horizontal line is considered vertical. Vertical line can be observed in the skyline or the edge of a book.
- Parallel Line: Parallel Lines are found in striped patterns or fence posts, are two lines on the same plane maintaining equal and constant distance without intersecting.
- Intersecting Line: Lines on the same plane intersecting at a specific point are known as Intersecting Lines. The image added below shows the parallel and intersecting lines, here l and m are parallel lines, and p and q are intersecting lines.
- Transversal Line: Transversal Line intersects two or more parallel or non-parallel lines at a defined point. Transversal line and parallel line should be lie on the same plane.
- Perpendicular Line: Perpendicular Lines occur when two lines intersect each other on the same plane, forming a right angle (90°) at the point of intersection. The perpendicular lines AB and CD are shown in the image added below.
- Secant Line: A line is considered a secant line to a circle when it intersects the circle precisely at two points.
- Skew Line: Skew Lines are lines lacking intersections with each other but unevenly spaced, distinct from parallel lines.
- Tangent Line: Tangent Lines are lines that touch the curve at only one point.
- Diagonal Line: Diagonal Line is a straight line that is set at an angle and connects the opposite corner of any shape.
- Contour Line: Contour Line defines the outer edges of any shape. For example- the outline of a mountain range.
- Dotted Line: Dotted Line are used in design to indicate a path or connection. For example- Dotted lines on a map.
General Equation of Line
General Equation of a Line in two variables with a degree of one is expressed as Ax + By + C = 0, where A and B cannot both be zero; and A, B, and C are real numbers.
This representation is commonly known as a linear equation in two variables.
For example: Consider the example 4x + 9y – 6 = 0, where A = 4, B = 9, and C = -6.
This equation is a specific example of the general form of line, showing the coefficients A, B, and C associated with the line.
Various Forms of Equation of Line
There is various ways to show the equation of line. These forms are:
Point-Slope form
In Point-slope form, the line is passing through a point (x0,y0) with a slope of m then the equation is expressed as:
(y – y0) = m(x – x0)
Two-Point form
In Two-point form, the line passes through two points say (x1,y1) and (x2,y2) then the equation of line is expressed as
y – y1 = {(y2-y1)/(x2-x1)}(x – x1)
Slope-Intercept form
In Slope-Intercept form, a line with a slope of m intersecting the y-axis at a distance of a point (0, c), that distance is known as the y-intercept of a line.
Equation of the line will be given as
y = mx+c
Intercept form
When a line intersects the x-axis at (a, 0) and the y-axis at (0, b), the equation can be derived using the two-point form,
y – y1 = {(y2-y1)/(x2-x1)}(x – x1)
By putting the value, we get
y-0 = {(b-0)/(0-a)}(x-a)
By rearranging the equation, we get
x/a+y/b = 1
This expression is recognized as the intercept form.
Also, Read
Examples on Line
Example 1: Determine the equation of the line for x-axis and y-axis.
Solution:
The x-axis is represented by the equation y=0, while the y-axis is denoted by x=0.
Example 2: Determine the equation of the line with a slope of 6 passing through the point (9,3).
Solution:
We have the slope m=6 and passing coordinates (x0,y0) = (9,3)
Now, by using
(y-y0) = m(x-x0)
(y-3) = 6 (x-9)
y-3 = 6x-54
6x-y = 51
y = 6x-51
Example 3: Determine the equation of a line when passing through (3,2) and (-4,8).
Solution:
Here, the Line is passing through coordinates (x1, x2) = (3,2) and (y1, y2) = (-4,8)
Now,
y-y1 = {(y2-y1)/(x2-x1)}(x-x1)
y-2 = {(8-2)/(-4-3)}/(x-3)
y-2= {6/-7}/(x-3)
-7y+14 = 6x-18
6x+7y-32=0
Example 4: Determine the y-intercept of a line 2x+5y=20.
Solution:
The given equation is 2x+5y=20
Change this equation in this form
x/a+y/b =1.
Now,
2x+5y = 20
2x/20 + 5y/20 = 1
x/10 + y/4 = 1
Hence, the y-intercept is 4.
Example 5: Determine the equation of a line passing through (1,4) and m=6.
Solution:
We have the slope m=6 and passing coordinates (x0,y0) = (1,4)
Now, by using
(y-y0) = m(x-x0)
y-4 = 6(x-1)
y-4 = 6x-6
y = 6x-2
Practice Questions on a Line
Q1. Determine the slope for the equation y = 2x + 3.
Q2. Find the point where the line 3x + 6y = 15 intersects the y-axis, means y-intercept.
Q3. Write the equation of a line when passing through the points (3,9) and (-2,4).
Q4. Find the equation of a line passing through the point (2,8) with a given slope of 4.
Q5. Determine the point where the line given by the equation y = 2x + 3 intersects the x-axis, means x-intercept.
FAQs on Line
What is the Definition of Line?
A Line is a one-dimensional figure with no endpoints.
What is the Definition of Ray?
In geometry, a Ray is a one-dimensional figure originating from a fixed point, extending infinitely in one direction without having an endpoint.
What is the Definition of Line Segment?
A Line Segment is a portion of a line characterized by two definite endpoints in both directions, possessing a defined length.
What is a Secant Line?
A Line is considered a secant line to a circle when it intersects the circle precisely at two points.
What is a Tangent Line?
Tangent Lines touch a circle at only one point.
What are Parallel Lines?
Two Lines are termed Parallel if they lie in the same plane maintaining equal and constant distance without intersecting.
What Are Perpendicular Lines?
Two lines are called perpendicular lines if they intersect each other at rith angles, i.e. at 90 degrees.