Parallel and Perpendicular Lines: Definition, Examples, Equations

If we look in our surroundings, we will find enormous examples of parallel and perpendicular lines. Some examples of parallel lines include railway tracks, opposite edges of the textbooks, and the opposite walls of the cubical room. Similarly, examples of perpendicular lines include the corners of a wall, the English alphabet L, the tiles in the kitchen, the hands of a clock when it struck exactly \(3\) o’clock, and the corners of your study table, to name a few. This article will discuss Parallel and Perpendicular Lines, their properties, and the construction steps.

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Parallel and Perpendicular Lines

Let us look at parallel and perpendicular lines and their characteristics one by one.

What are Parallel Lines?

Two lines at an equal distance from each other that never meet are said to be parallel lines.

Two or more lines that are at the same distance apart, never merging or diverging are called parallel lines.

line \(a\) and line \(b\) are parallel lines in the above figure. 

Parallel lines \(a\) and \(b\) are symbolically written as \(a{\rm{||}}b{\rm{ }}\)

Examples of Parallel Lines

We can find so many examples for parallel lines from things we come across in our daily life.

The opposite edges of a postcard are parallel to each other.

In the above figure, the opposite vertical lines of the railway tracks are parallel to each other. Similarly, the horizontal lines are also parallel to each other. Two lines that are perpendicular to each other are said to be parallel.

Construction of Parallel Lines

To construct parallel lines, we require a ruler and a compass. The following steps are followed to construct parallel lines,

1. First, draw a line \(m\) with the help of a ruler.

2. Mark a point \(X\) on a line \(m\), as shown in the figure.

3. Choose the point \(Y\) away from the line \(m\) and join the points \(X\) and \(Y\) as shown in the figure.

4. Consider \(X\) as the centre and any suitable radius draw an arc cutting the line segments \(XY\) at a point \(V\) and the line \(m\) at a point \(Z\).

5. With \(Y\) as the centre and a radius of \(XZ\) or \(XV\), draw an arc cutting the line segments \(XY\) at a point \(U\) as shown below.

6. With \(U\) a centre and radius equal to \(ZV\) cut an arc \(PQ\) at a point \(A\). Draw a line that connects the points now \(A\) and \(Y\). Then, name the obtained line as \(n\) so that the line \(m\) is parallel to line \(n\), i.e., \(m∥n\).

Perpendicular Lines

When two lines intersect at a right angle, the lines are said to be perpendicular.

When two lines (or rays or segments) intersect, the angles generated between them are right angles. The lines \(l\) and \(m\) in the diagram below are perpendicular lines. Symbolically it is written as \(m\, \bot\, l\)

Examples of Perpendicular Lines

We can give many examples for perpendicular lines (or line segments) from things we come across in our daily life.

In the above figures, the edges of the set square and blackboard, the arms of the clocks at \(3\) o’clock are examples of perpendicular lines. The angle formed between the perpendicular lines is exactly \({90^ \circ }\).

The edges of a postcard meet and form an angle that is the right angle.

Perpendicular Line Through a Point

On the tracing paper, draw a line \(l\). On this line, make a point \(P\). Now draw a perpendicular across \(P\) on \(l\). At point \(P\), fold the paper so that both sides of the fold are parallel. We discover that the crease on the paper is perpendicular to the line \(l\) on it when we unfold it.

Perpendicular Line to a Line Through a Point which is Not on It

The construction steps are listed below.

1. Draw a line \(l\) with a point \(A\) on the outside.

2. Draw an arc that cuts the provided line \(l\) at two unique locations, \(M\) and \(N\), using \(A\) as the centre.

3. Draw two arcs that intersect at a point say \(B\), on the side opposite to point \(A\) on line \(l\), using the same radius (or any other radius) with \(M\) and \(N\) as centres.

4. Connect the points \(A\) and \(B\) now. A perpendicular line to the given line \(l\) is line \(AB\).

Perpendicular Bisector

A perpendicular bisector of a line segment divides it into two equal portions and is perpendicular to it.

Construction of Perpendicular Bisector of a Given Line Segment: Method of Ruler and Compass

Steps of Construction:

1. Draw a line segment \(AB\).

2. Then, set the compass with a radius more than half of the length of the line segment \(AB\). With \(A\) as the focal point, draw two arcs, one below and the other above the line segment \(AB\).

3. Draw two arcs with the same radius and \(B\) as the centre to cut the previous arcs at points \(M\) and \(N\), respectively.

4. Finally, join the points \(M\) and \(N\). So, the perpendicular bisector of the line segment \(AB\) is line \(l\). At point \(P\), line \(l\) intersects line segment \(AB\).

Properties of Perpendicular and Parallel Lines

Parallel Lines	Perpendicular Lines
The two non-vertical lines that are in the same plane and have the same slope are said to be parallel.	When the two non-vertical lines in the same plane intersect at a right angle, they are perpendicular.
Two parallel lines never intersect each other.	Perpendicular lines meet at right angles to each other.

Parallel and Perpendicular Line Equations

If the equation of any line is given by \(ax+by+c=0\), then a line parallel to the provided line’s equation is represented as \(ax+by+d=0\), where \(d\) is a constant.

A line perpendicular to a given line’s equation \(ax + by + c = 0\) is  \(bx – ay +d = 0\), where \(d\) is a constant. 

Parallel and Perpendicular Lines Slope

Slope of a Line

The slope or gradient of a line is represented by \(m = \tan \theta ,\) where \(\theta \) is the angle made by the line with the \(x\)-axis.

If the coordinates of two points are given as \(\left( {{x_1},{y_1}} \right)\) and \(\left( {{x_2},{y_2}} \right)\) then m is given by \(m = \frac{{{y_2} – {y_1}}}{{{x_2} – {x_1}}}\).

Slope of Two Parallel Lines

Two parallel lines have the same slope. Consider \({m_1}\) and \({m_2}\) are the two slopes of two parallel lines, then \({m_1} = {m_2}\).

If two lines have the same slope, they are said to be parallel.

Slope of Two Perpendicular Lines

The slopes of two perpendicular lines are multiplied, and the result is \(-1\).
If the slopes of two perpendicular lines are \({m_1}\) and \({m_2}\), then \({m_1} \times {m_2} = \,- 1\)

Parallel and Perpendicular Lines in Co-ordinate Geometry

If two lines in the same plane have the same slope, they are parallel. The two lines in the same plane intersecting at a right angle are perpendicular; horizontal and vertical lines are perpendicular. They form the axes of the coordinate plane.

Two parallel lines in the Cartesian coordinate plane are shown below.

Two perpendicular lines in the Cartesian coordinate plane are shown below.

Learn Concept of Lines in Geometry

Solved Examples – Parallel and Perpendicular Lines

Q.1. Find the slope of the line parallel to the line \(4x – 5y = 12\).
Ans: We know that to find the slope of the line \(4x – 5y = 12\), We needed to convert the string to \(y = mx + b\) which is the slope-intercept form. It means we need to need to solve for \(y\).
\(4x – 5y = 12 \Rightarrow 5y = 4x – 12 \Rightarrow y = \frac{4}{5}x – \frac{{12}}{5}\)
The slope of the line \(4x – 5y = 12\) is  \(m = \frac{4}{5}\).
As a result, the slope of each line parallel to the line \(4x – 5y = 12\) will be \(m = \frac{4}{5}\).

Q.2. Find the equation of a line that is parallel to \(y =\, – 3x + 5\) and passing through the point \((2, -7)\).
Ans: The slope of the line \(y = \,- 3x + 5\)  is \(m =\, – 3\) So, the slope of the line parallel to this line is also \(-3\).  We will use the point \((2,-7)\) from which a line must pass through.
Substitute these values into the Point-Slope Form and solve for \(“y”\).
\(y – {y_1} = m\left( {x – {x_1}} \right)\)
\( \Rightarrow y – ( – 7) = \,- 3(x – 2)\)
\( \Rightarrow y + 7 =\, – 3(x – 2)\)
\( \Rightarrow y + 7 = -\, 3x + 6\)
\( \Rightarrow y = \,- 3x – 1\)
Therefore, the equation of a line that is parallel to \(y =\, – 3x + 5\) is \(y = \,- 3x – 1\).

Q.3. Determine whether the lines are parallel or perpendicular. \(3x + 4y = 2\) and \(8x – 6y = 5\).
Ans: Writing each equation in the slope-intercept form:
\(3x + 4y = 2 \Rightarrow 4y = – 3x + 2 \Rightarrow y = – \frac{3}{4}x + \frac{1}{2} \Rightarrow {\rm{ slope }} = \frac{{ – 3}}{4}\)
\(8x – 6y = 5 \Rightarrow 8x – 5 = 6y \Rightarrow y = \frac{4}{3}x – \frac{5}{6} \Rightarrow {\rm{ slope }} = \frac{4}{3}\)
The slopes are the inverses of each other.
Therefore, the lines are perpendicular.

Q.4. Check whether the lines are parallel or perpendicular. \(7y + 1 = 7x\) and \(x + 5 = y\)
Ans: Writing each equation in the slope-intercept form:
\(7y + 1 = 7x \Rightarrow 7y = 7x – 1 \Rightarrow y = x – \frac{1}{7} \Rightarrow \,{\text{slope}} = 1\)
\(x + 5 = y \Rightarrow y = x + 5 \Rightarrow \,{\text{slope}} = 1\)
The slopes are the same.
Therefore, the lines are parallel.

Q.5. Find the equation of a line that is parallel to \(y = 2x + 1\) and passing through the point \((5, 4)\).
Ans: The slope of the line \(y = 2x + 1\)  is \(m=2\). So, the required line also has a slope equal to \(2\). We will use as the point \((5, 4)\) where this unknown line must pass through.
Substitute these values into the Point-Slope Form and solve for \(“y”\).
\(y – {y_1} = m\left( {x – {x_1}} \right)\)
\( \Rightarrow y – 4 = 2(x – 5)\)
\( \Rightarrow y – 4 = 2(x – 5)\)
\( \Rightarrow y – 4 = 2x – 10\)
\( \Rightarrow y = 2x – 6\)
Therefore, the equation of a line that is parallel to \(y = 2x + 1\) is \(y = 2x – 6\).

Summary

Parallel lines are made up of two or more lines with the exact distance between them and never merge or diverge. If two lines in the same plane cross, they are said to intersect at a right angle, it is called a right angle intersection, and they are said to be perpendicular. We studied the definitions, constructions, properties, and examples of parallel and perpendicular lines.

We also looked at the equation of parallel and perpendicular lines and slopes. 

Frequently Asked Questions (FAQs) – Parallel and Perpendicular Lines

The most frequently asked questions about parallel and perpendicular lines are answered here:

We hope this detailed article on parallel and perpendicular lines helped you in your studies. If you have any doubts or queries regarding this topic, feel to ask us in the comment section and we will be more than happy to assist you.