A line in geometry is a straight path that goes on forever in both directions. It has no thickness and is usually drawn between two points, but it keeps going without stopping. Lines are important for making shapes, measuring distances, and understanding angles. For example, the edge of a ruler can represent 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.
Line, Line Segment and Ray
They sound and look similar but have significant difference between them.
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.
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.
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.
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.
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.
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.
Bisecting Lines
Bisecting lines are those lines that divide a line segment into two equal parts. That object can be an angle, triangle, any polygon or a line segment. They pass through the midpoint of the object. Bisecting lines are commonly used in geometry to divide angles or line segments equally.
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.
Few more types of Lines
- 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.
Real-World Applications of Lines in Geometry
Lines in geometry are not just abstract concepts; they have many real-world applications. In architecture and engineering, lines are used to design buildings, bridges, and other structures. For example, architects use parallel and perpendicular lines to create strong and balanced structures. Similarly, in art, lines are
Solved Examples
Example 1: Determine the equation of the line for x-axis and y-axis.
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).
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).
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.
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.
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
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.