Horizontal Line

Horizontal lines are defined as lines that are parallel to the horizon or the ground, hence the name horizontal line. Horizontal line has zero slope, i.e. the inclination angle of these lines is zero degrees. If the horizontal lines are drawn on the Cartesian planes they only make intercept along the y-axis as they are always parallel to the x-axis and never cut it.

In this article, we will learn about the horizontal line, its properties, the slope of the horizontal line, the equation of the horizontal line, examples and FAQs related to the horizontal lines, and others in detail.

Horizontal Line Definition

We know that a line is a straight path that joins two or more two points and runs up to infinity. Thus, we define horizontal lines as lines that are parallel to the ground or the Horizon and are at a constant height from the ground. If we plot these lines on the Cartesian system then these lines are the lines that have no intercept on the x-axis but have an intercept on the y-axis. These lines have zero inclination, i.e. they have zero angle with the x-axis or the ground.

These lines form the base of the various objects, shapes, and figures, that we study in Geometry. Suppose we have to draw a rectangle, square, triangle, trapezium, etc then the base of these figures are mostly horizontal lines. We also observe horizontal lies in real life as the lines on the floor and the roof of the rooms, the base of the stairs, etc are also made by the horizontal lines.

These lines are also known as sleeping lines as they have no vertical movement and always remain at a constant height from the ground. The image added below shows the horizontal line.

Here, in the figure added above lines l and m are Horizontal lines.

Slope of Horizontal Line

We have already mentioned that the slope of the horizontal line is zero. Now let’s learn how the slope of the horizontal line is zero. We calculate the slope of a horizontal line by using the formula,

Slope = Rise/Run

OR

Slope of Line = Change in y-coordinate/Change in x-coordinate

Where Rise is the height gained by the line while running left to right, as we already know that the horizontal line is parallel to the x-axis and is always at the constant height thus, we say that these lines have zero rise so the slope of these lines is,

Thus, Slope = 0/Run = 0

Thus, it is concluded that the slope of the horizontal line is zero.

Drawing a Horizontal Line

Horizontal lines are easily drawn by using the steps discussed below,

Step 1: Take a point on the Cartesian plane for which we have to find the horizontal line. Suppose the point is (1, 2)

Step 2: Marks the y-coordinate of the point. In this case, the y-coordinate is 2.

Step 3: Mark other points where the y-coordinate is the same as the point in Step 1. Let the other points are (-2, 2), (0, 2), and (7, 2)

Step 4: Join all the points to get a line segment and extend them on both sides to get a horizontal line.

This is the required horizontal line passing through the point (1, 2) and has a slope of zero.

Horizontal Line Equation

We know that the equation of the line in the 2-D coordinate system is,

y = mx + c

Where,

• m is the slope of the line
• c is the intercept on the y-axis

We know that for the horizontal line, the slope is zero. Substituting this value in the above equation we get the equation of the horizontal line to be,

y = 0x + c

y = c

Where c is a constant.

Thus, the above equation y = c is the equation of the horizontal line. This equation signifies that the horizontal line is a line that passes through all the points in the Cartesian where the y coordinate is equal to ‘c’. This line cut has no x coordinate and hence this line never cuts the x-axis and it cuts the y-axis at point (0, c).

Thus, we can say that the equation of the horizontal line is, y = c(constant) and it passes through the point (a, c) where a can take any value and c is always constant.

Horizontal Line Test

A test that is used to define whether a function is a one-to-one function or not is the Horizontal line test. In the horizontal line test, we draw a horizontal line passing through any one point on the function and if the lines cut the function at any other point then the function is NOT a one-to-one function. Thus, for a function to be one-to-one it has to pass the Horizontal line test, i.e. any horizontal line must cut the function only once.

We know that one-to-one functions are the functions where for each value of x we have only one value of y. So if the Horizontal line passes through the function and cuts it only once then we can say that for the unique value of the y, we have a unique value of x. But if the horizontal line cuts the function more than once then we get two values for the unique value of y which is not the case for the one-to-one function.

Horizontal Line Test helps us to determine whether a function is a One-One Function. This can be understood with the help of the image added below.

In the first image, the function is one-to-one because the horizontal line passes through only one point of the function.

In the second image, the function is NOT one-to-one as the horizontal line passes through more than one point of the function.

Horizontal and Vertical Lines

Horizontal lines are lines that are parallel to the ground or the horizon. These lines are also called the sleeping lines. In the cartesian system, these lines are parallel to the x-axis, while for the Vertical lines, these are the lines that are perpendicular to the horizontal lines, they are called the standing lines. and are parallel to the y-axis in the cartesian system.

Horizontal lines are the lines that run left-to-right in the cartesian system whereas vertical lines are the lines that run up and down in the cartesian system.

Vertical and Horizontal Lines are perpendicular to each other. The image added below shows a Vertical and Horizontal Line.

The differences between vertical lines and horizontal lines can be easily understood by studying the table added below.

Horizontal Line	Vertical Line
These lines are parallel to the ground or the horizon.	These lines are perpendicular to the ground or the horizon.
The slope of the horizontal line is zero.	The slope of the vertical line is undefined.
Horizontal line made an angle of zero degrees with the horizon.	Vertical line made an angle of 90 degrees with the horizon.
Equation of the horizontal line passing through the point (h, k) is, y = k	Equation of the vertical line passing through the point (h, k) is, x = h
Horizontal lines are parallel to the x-axis in the Cartesian system.	Vertical lines are parallel to the y-axis in the Cartesian system.
Examples representing the horizontal lines are, Straight Road Bottom of Staircase Base of any Figure, etc.	Examples representing the vertical lines are, A Long Tower A Flagpole Height of any Building

Short note on horizontal line

A horizontal line in mathematics is perfectly level, parallel to the horizon. It runs from left to right and has a slope of 0. In geometry, it’s represented as a straight line connecting any two points at the same height on a plane. The equation for a horizontal line is of the form (y = k), where (k) is a constant value representing the height of the line on the y-axis.

Read More:

• Types of Lines
• Parallel Lines

Horizontal Line Examples

Example 1: Find the equation of the horizontal line passing through the point (1, -1).

Solution:

We know that the slope of the horizontal line is m = 0.

Given point (1, -1)

Equation of the line passing through a point (x₁, y₁) and having a slope (m) is,

y – y₁ = m(x – x₁)

Substituting the values in the above equation we get,

y – (-1) = 0(x – 1)

y + 1 = 0

y = -1

Thus, equation of the horizontal line passing through the point (1, -1) is, y = -1

Example 2: Find the equation of the horizontal line passing through the point (5, 9).

Solution:

We know that the slope of the horizontal line is m = 0.

Given point (5, 9)

Equation of the line passing through a point (x₁, y₁) and having a slope (m) is,

y – y₁ = m(x – x₁)

Substituting the values in the above equation we get,

y – (9) = 0(x – 5)

y – 9 = 0

y = 9

Thus, equation of the horizontal line passing through the point (5, 9) is, y = 9

Example 3: Find the equation of the horizontal line when the y-intercept of the line is 5.

Solution:

Equation of the horizontal line is,

y = k

where k is y-intercept

Given

• k = 5

Equation of the horizontal line,

y = 5

Thus, the equation horizontal line with y-intercept as 5 is, y = 5

Example 4: Find the equation of the horizontal line when the y-intercept of the line is -11/3.

Solution:

Equation of the horizontal line is,

y = k

where k is y-intercept

Given

• k = -11/3

Equation of the horizontal line,

y = -11/3

3y = -11

3y + 11 = 0

Thus, the equation horizontal line with y-intercept as -11/3 is, 3y + 11 = 0

FAQs on Horizontal Lines

1. What are Horizontal Lines?

Horizontal lines are lines that are parallel to the horizon or the ground. In the cartesian system horizontal lines are parallel to the x-axis.

2. What is the Equation of the Horizontal Line?

Equation of the horizontal line is,

y = k

where k is the intercept on the y-axis.

3. What is the slope of a horizontal line?

The slope of the horizontal line is always equal to zero, as they make zero degrees angle with the x-axis.

4. What are Examples of Horizontal Lines?

Examples representing the horizontal lines are,

• Straight Road
• Bottom of Staircase
• Base of any Figure, etc.

5. What are the Horizontal Lines on the Globe called?

Horizontal lines running on the globe are called Latitudes and they run parallel to the Equator.

6. What are the Properties of Horizontal Lines?

Various properties of the Horizontal lines are,

• They are parallel to the ground, horizon, and x-axis.
• They are perpendicular to the y-axis.
• The slope of the horizontal line is zero, etc.