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The horizontal line, in coordinate geometry, is the line parallel to the x-axis. And the line parallel to the y-axis is called the vertical line. The horizontal line will be perpendicular to the y-axis and the vertical lines will be perpendicular to the x-axis. Thus, horizontal and vertical lines are perpendicular to each other when drawn with at least one common point.
Also read: Slope-Intercept Form
Key Terms: Horizontal line and Vertical Lines, Horizontal Line, Straight Lines, Perpendicular
What is a Horizontal Line?
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Horizontal line can be described as a line which has the same y-coordinates. The slope of a horizontal line is always zero and doesn’t change when we move from different ends of the line.
It is the line parallel to the x-axis if the base is considered. So we can say that the horizontal line is the straight line parallel to the x-axis and travelling the direction sideways or left to right without deviating or travelling in an upward or downward direction.
Equation of the Horizontal Line
Horizontal lines remain parallel to the x-axis in all cases. Horizontal lines never intersect the x-axis, they only intersect the y-axis. For horizontal lines, the value of x can change but the value of y always tends to be constant.
When we consider the intercept equation of slope, by using ‘m=0’ in ‘mx+b’ because the value of the slope is taken 0 for horizontal lines. Then, the equation becomes y=b. In such a case, the ‘b’ is taken as the real number and it is considered the y-intercept for the y-coordinate.
Horizontal Line Symmetry
Horizontal Line symmetry is the orientation either vertical, horizontal, or diagonally placed. Suppose that a line of symmetry for any given figure is parallel to the horizontal plane then, the line will be determined as the horizontal symmetry of the line.
Horizontal Line Test
Horizontal line test is used to determine whether any function has a one-to-one derivative or not. For this, a linear function is used. Let f[x] and assume we need to find its inverse function. So we need to show that f[x] is actually a one-to-one function. For this, we need to represent that it passes through the horizontal test.
Hence, f[x] will be proven as a one-to-one function only if it crosses the horizontal line at a single point only. But if it touches the horizontal line at more than one point, it is not a one-to-one function. So the horizontal line test helps us to determine if the f[x] has a one-on-one function or not by seeing whether they have an inverse function or not.
| Note: The midpoint of the line is the point at which it gets divided into two equal congruent parts by a point of the segment. |
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What is a Vertical Line?
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Vertical line is the line in a coordinate plane that represents the direction from up to down. This line is always parallel to the y-axis and its x-coordinate is always constant.
The y-coordinate of the vertical lines can have two values, while its x-coordinate always has the same value. The vertical line also has zero slopes. So, we can say that a vertical line is a straight line in the direction of the y-axis which goes upward in or downward in the coordinate plane but does not travel on adjacent sides.
Vertical Line
Equation of Vertical Line
If we consider a vertical line L is at a distance “b” from the y-axis where, the ordinate of every point lying on the line is either b or – b. Thus, the equation of the line L will be either x = b or x = – b. However, the sign will depend upon the position of the line that is whether the line is on the left or right side of the y-axis.
Vertical Line Symmetry
Vertical Line symmetry occurs when the line of symmetry is parallel to the vertical plane. Both types of symmetry can be shown in different shapes. These lines divide the shapes into two parts such that they are mirror images of one another.
Characteristics of Vertical LineThe characteristics of a vertical line are:
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Also read: Properties of Parallel Lines
Important Formula of Horizontal and Vertical Lines
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Here are all the important formulas of the horizontal and vertical lines:
- The equation of the straight line is y = mx + c [here, m denotes the slope while b denotes the y-intercept].
- The equation of the horizontal line is y=b [where ‘m= 0’ in ‘mx + b’ in the straight line equation].
- The formula for the slope of the vertical line is m = y2 – y1/x2 – x1[ where x1 and y1 are the coordinates of the first line while the x1 and x2 are the coordinates of the second line].
- The equation of the vertical line is x = a [where a= it is the point where this line intersects the x-axis and x= respective coordinates of any point lying on the line].
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Graphing Horizontal and Vertical Lines
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A line is a geometric tool used to describe entities with negligible dimensions. It can be denoted by two points or by a single alphabet. For representing the line an equation can be used:
| y = mx + c (here, m denotes the slope, and b denotes the y-intercept) |
A Cartesian coordinate system when incorporated with a plane describes every point by a set of numerical coordinates. When we consider a Cartesian coordinate system in describing the lines then, the lines are denoted according to the axes.
Axes are the two lines intersecting at 90 degrees to each other. These numerical coordinates denote the distance from the fixed two perpendicular coordinate axes. With respect to the Cartesian coordinate system, lines can be of two types:
- Horizontal Lines: This line travels in the direction of east to west.
- Vertical Lines: This line travels in the direction of the north to south.
As we already know, that vertical line has an up and down direction. They radiate from a particular point of the x-axis. If we want to draw a vertical line in a graph on an x-plane, we need to make a line from that point that will be straight in formation.
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Things To Remember
- In coordinate geometry, the horizontal line is described as the line which has the same y-coordinates.
- The slope of a horizontal line is always zero and doesn’t change when we move from different ends of the line.
- The line of symmetry can be discovered in various shapes in any given figure.
- Vertical line is the line in a coordinate plane that represents the direction from up to down or we can say it goes in that direction.
- The equation of the straight line is y = mx+c [here, m denotes the slope while b denotes the y-intercept].
- The equation of the horizontal line is y = b [where ‘m=0’ in ‘mx+b’ in the straight line equation].
- The formula for the slope of the vertical line is m = y2-y1/x2-x1[ where x1 and y1 are the coordinates of the first line while the x1 and x2 are the coordinates of the second line].
- The equation of the vertical line is x = a [ where a= it is the point where this line intersects the x-axis and x= respective coordinates of any point lying on the line].
Also Read: NCERT Solutions for Class 6 to 12 PDFs
Sample Questions
Ques. Determine the slope for the mentioned coordinate points- [6,2] and [6,0]. (2 marks)
Ans: We will take the following approach to solve this question:
It is given in the question that y1= 0 while y2= 2 and x1= 6 while x2= 6.
We know that the formula for the slope of the vertical line is: m = y2-y1/x2-x1.
From the formula we know that the slope will be,
m= 2 – 0/6 – 6 = 2/0, as the denominator is 0 the slope ‘m’ remains undefined.
This also represents that the slope of the vertical line remains undefined.
Ques. Determine the slope of the mentioned coordinate points – [7,3] and [7,0]. (3 marks)
Ans: We will take the following approach to solve this question;
It is given in the question that y1= 3 while y2= 0 and x1= 7 while x2= 7.
We know that the formula for the slope of the vertical line is: m = y2-y1/x2-x1.
From the formula we know that the slope will be,
m= 3 – 0/7 – 7 = 3/0, as the denominator is 0 the slope ‘m’ remains undefined.
This also represents that the slope of the vertical line remains undefined.
Ques. Find the horizontal line equation of a given straight line whose y-intercept is given as the [0,2]. (2 marks)
Ans: It is given that the y-intercept is 0 and 2.
We are already aware that the equation of the horizontal line is y=b [where ‘m=0’ in ‘mx+b’ in the straight line equation].
It is given that the value of the y-intercept is given as 2 which means b=2.
That’s why we can say that the equation of the horizontal line is y=2.
Ques. Assume that the equation of a line is y = 3 then, tell whether it is horizontal or vertical. (2 marks)
Ans: It is given that the line of the equation has y=3 this makes it a horizontal line equation because the line will intersect the y-axis at point y=3 and further it will remain parallel to the x-axis. Hence, we can say that the given line is a horizontal line parallel to the x-axis.
Ques. Give an example of a vertical line equation representation. (1 mark)
Ans: The definition of the vertical line is that it intersects the x-axis and always travels parallel to the y-axis. So as an example of the vertical line we can give an equation x=-4.
Ques. Give the definition of the horizontal line. (1 mark)
Ans: A line that has a constant or same y-coordinate in the coordinate plane and which travels parallel to the x-axis is called a horizontal line. It only travels in the left to right direction.
Ques. Determine an equation assuming that a horizontal line passes through the point (4,7). (3 marks)
Ans. Assuming the equation, the straight line in the slope-intercept forms
⇒y = mx + b
Now, we know that slope of the horizontal line = 0
⇒ m = 0
Hence, for a line to pass through (4,7), the equation = y = (0) x + b
The line cuts the respective y-axis at points (0,7).
Therefore, y-intercept = 7.
Thus b = 7
Horizontal line is y = (0)x + 7 ⇒ y = 0 +7 = 7
Ques. Is y = 0 a horizontal line? (1 mark)
Ans. A horizontal line can possess various values of x, but can only have one possible value for y. Hence, y is the form of “y = 0”.Thus, y = 0 is a horizontal line.
Ques. Show the equation for the horizontal line with points (7, 10). (1 mark)
Ans. As we are aware, the equation of a horizontal line is y = b.
Here, b is known as the coordinate of the point (a, b).
Thus, the equation for it includes the point (7, 10) where y = 10.
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