Vertical lines are basic concepts in geometry and coordinate systems, running straight up and down without any inclination. They are perpendicular to the base of any geometrical object and generally, we state the bottom of the object as a base. In context to the cartesian coordinate system, vertical lines are defined as lines that are parallel to the y-axis or perpendicular to the x-axis.
This article discusses vertical lines in detail, covering their definition, diagrams, and their relationship with other lines, providing useful insights for solving geometrical problems.
Vertical Lines Definition
Vertical lines are lines in which all points have the same x-coordinate. In other words, a line that is perpendicular to the x-axis and parallel to the y-axis is called a Vertical Line.
In real life, we observe various examples of vertical lines such as a long tower, the legs of a table and a chair, a long tree, etc. The slope of the vertical line is undefined as it makes a 90° angle with the x-axis. The vertical line goes from top to bottom in the Cartesian plane.
Vertical Line on Coordinate Plane
On the Coordinate Plane any line parallel to the y-axis is called the vertical line. A vertical line has a fixed x-coordinate and a variable y-coordinate. The general point on a vertical line is (c, b) where c is constant and the value of b changes accordingly.
At the x-axis, the vertical line has the coordinate (c, 0) and it is perpendicular to the x-axis.
The image added below shows the vertical line,
Equation of Vertical Line
The equation of the vertical line is given as,
x = h
Where h is any real constant.
All the points in these lines have the coordinates as (h, b) where the h is constant and b is variable.
For the equation of the vertical line i.e., x = 11. We can say that this line passes through the point (11, 0) on the x-axis and various points on this vertical line are, (11, -3) and (11, 9), (11, 8/9), etc.
Vertical Line Example
There are various scenarios where we see vertical lines in real life and some of them are we see vertical lines at the corner of the building these lines run across the height of the building, and the height of a tree or a mountain is measured using a vertical line.
In coordinate geometry, various examples of the vertical line are, x = 9 is a vertical line passing through points (9, 0), (9, -1), (9, -2), (9, 4), (9, 8), etc. This line cuts the x-axis at (9, 0) and is parallel to the y-axis.
Some other examples of vertical lines are,
- Equation of a vertical line through (-1, 0) is x = -1
- Equation of a vertical line through (8, 0) is x = 8
Slope of a Vertical Line
The slope of a vertical line is undefined. This can be understood by the definition of the slope of the line as,
Slope of a line = Change in y-coordinate/ Change in x-coordinate
OR
m = (y2 - y1) / (x2 - x1)
For a vertical line, we know that the x-coordinate never changes, and thus x2 = x1 = x, so x2 - x1 = 0
⇒ m = (y2 - y1)/0 = Undefined
Thus, the slope of the vertical line is undefined.
Properties of Vertical Lines
Vertical lines have some special properties that include,
- Vertical line is a line that is always parallel to the y-axis.
- Vertical line is a line that is always perpendicular to the x-axis.
- The slope of the vertical line is undefined or infinity.
- Equation of the vertical line is of the form, x = h(constant).
- The value of x coordinate remains constant for every point on the vertical line.
Difference Between Horizontal Line and Vertical Line
There are key differences between both vertical lines and horizontal lines, some of these differences are listed in the following table:
| Aspect | Vertical Line | Horizontal Line |
|---|
| Orientation | Parallel to the y-axis | Parallel to the x-axis |
|---|
| Slope | Slope is undefined or infinite | Slope is zero |
|---|
| Equation | Equation of vertical line is,
x = constant value | Equation of horizontal line is,
y = constant value |
|---|
| Direction | Extends from top to bottom | Extends from left to right |
|---|
| Intersection | Intersects the x-axis at one point | Intersects the y-axis at one point |
|---|
| Graph | Graph of a vertical line is a straight line parallel to the Y-axis | Graph of a horizontal line is a straight line parallel to the X-axis |
|---|
| Example | Equation of vertical lines is,
x = -2
x = 7, etc | Equation of horizontal lines is,
y = 3
y = -6, etc |
|---|
| Angle to X-axis | 90° | 0° |
|---|
| Angle to Y-axis | 0° | 90° |
|---|
Vertical Line of Symmetry
A line running from the top to the bottom of any figure that divides the figure into two identical halves that are mirror images of each other is called the vertical line of symmetry.
There are various figures in which the vertical line of symmetry is observed that include square, rectangle, circle, etc. The image added below shows the vertical line of symmetry of these figures,
A part of these the alphabet in the English language also shows a vertical line of symmetry. There are a total of 11 alphabets in the English language that shows the Vertical line of symmetry that includes, A H I M O T U V W X Y. The image added below shows the vertical line of symmetry for the same.
Read More about Symmetry.
Vertical Lines Solved Examples
Example 1: Find the equation of the vertical line passing through the point (1, -1).
Solution:
Given point (1, -1)
Equation of the vertical line passing through a point (h, k)
x = h
Substituting the values in the above equation we get,
x = 1
Thus, equation of the vertical line passing through the point (1, -1) is x = 1.
Example 2: Find the equation of the vertical line passing through the point (5, 9).
Solution:
Given point (5, 9)
Equation of the vertical line passing through a point (h, k)
x = h
Substituting the values in the above equation we get,
x = 5
Thus, equation of the vertical line passing through the point (5, 9) is x = 9.
Example 3: Find the equation of the vertical line when the x-intercept of the line is 5.
Solution:
Equation of the vertical line is,
x = h
where h is x-intercept
Given
Equation of the vertical line,
x = 5
Thus, the equation vertical line with x-intercept as 5 is, x = 5
Example 4: Find the equation of the vertical line when the x-intercept of the line is -11/3.
Solution:
Equation of the vertical line is,
y = k
where h is x-intercept
Given
Equation of the vertical line,
x = -11/3
3x = -11
3x + 11 = 0
Thus, the equation vertical line with x-intercept as -11/3 is, 3x + 11 = 0
Read More,
Conclusion
All the basic of the vertical lines can be summarized as,
- Vertical line is a straight line perpendicular to the baseline or the horizontal line.
- Equation of a vertical line is x = h (constant).
- A vertical line is always parallel to the y-axis and perpendicular to x-axis.
- Slope of vertical lines is not defined.
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