Which step is the same in the construction of parallel lines in the construction of a perpendicular line through a point on the line?

• Written By Keerthi Kulkarni
• Last Modified 20-10-2022

Lines are one-dimensional figures which have lengths only. The lines that are non-intersecting or not touching each other and having the same distance between them in all points are called parallel lines. The lines that are intersecting at right angles with each other are known as perpendicular lines. Generally, lines are constructed by using the ruler and compass, etc. This article will study different methods commonly used for the Construction of Parallel and Perpendicular Lines in detail with all steps. We have also provided examples to help you understand the techniques and concepts in detail. Read on to learn more about the construction of parallel and perpendicular lines.

Parallel and Perpendicular Lines

Let us begin by understanding what parallel and perpendicular lines are using diagrams and examples.

Parallel Lines

Parallel lines are lines, which never intersect or meet each other at any point. Parallel lines lie on the same plane. The distance between parallel lines is always the same. That means the parallel lines are always equidistant from each other. The symbol for “parallel to” is “\(∥\)”.

Here, the lines \(A\) and \(B\) are parallel lines, as they are not intersecting.

Perpendicular Lines

Two intersecting lines are said to be perpendicular lines if they intersect each other at an angle of right angle \(\left(90^{\circ}\right)\). Thus, the angle formed between two perpendicular lines is \(90^{\circ}\).

The symbol used to represent the perpendicular lines is “\(\perp\)”.

In the below figure, the lines \(X Y\) and \(A B\) are perpendicular to each other. As the angle between them is \(90^{\circ}\). Mathematically, they are represented as \(X Y \perp A B\).

In geometry, all the figures are constructed by using some methods and steps. The various tools used to construct the geometrical shapes are listed below:

1. Scale or Ruler: It is used to draw the line segment. The ruler is used to measure the length of the sides of the plane figures and the line segments
2. Protractor: Protractor is the tools used to measure or draw the angle of any measure. 
3. Compass: It is the basic tool used to draw the arcs.

Two lines that intersect at an angle of right \(\left(90^{\circ}\right)\) are called the perpendicular lines. The steps of constructing the perpendicular lines are given below:

Step-1: Draw one line segment \((PQ)\) of any length by using the ruler.

Step-2: Place any point \((R)\), not on the line segment.

Step-3: By keeping the tip of the compass at the point and taking any radius, draw an arc that intersects the given line segment at \((X, Y)\).

Step-4: Draw circles at the intersection points \((X, Y)\) of arc and line as shown by using the compass with the radius equal to more than half of the length of the line segment \((XY)\) between the intersecting points.

Step-5: Draw the line through the intersection points of the two circles \((A, B)\) and the given point as shown below:

Here, the blue colour lines \((PQ, RS)\) shown in the figure are the perpendicular lines perpendicular to each other.

Construction of Perpendicular from a Point on It

The steps of construction of the perpendicular line to the given line from a point on it are given below:

Step-1: Consider any line \(P Q\) and point \(M\) on it.
Step-2: Draw arcs on either side of the point \(M\) on the line \(P Q\) using a compass of any radius. Mark the intersection points as \(A\) and \(B\).
Step-3: By taking the radius equals more than half of the line segment \(A B\), draw arcs from point \(A\) above the line.
Step-4: With the same radius, draw an arc from point \(B\) above the line.
Step-5: Mark the intersection points of arc as \(C\). Now join \(C\) and \(M\).

Thus the line drawn from the points \(C\) and \(M\) gives the required perpendicular line drawn to the given line from the given point.

The perpendicular bisector of a line is the line drawn at the right angles to the given line, dividing the given line into two equal parts. The steps of construction of a perpendicular bisector to the given line are listed below:

Step-1: Draw a line \(AB\) of any length by using the ruler.
Step-2: By taking the compass of radius equals to more than half of the length of the given line and placing the tip of the compass at one point of the line (say \(A\)), draw an arc above and below the given line.
Similarly, from the other point (say \(B\)), draw another arc that intersects the previous arc by using the compass of the same radius above and below the given line.

Step 4: Join the intersection points of arcs \(C\) and \(D\) and extend the line drawn, which gives the perpendicular bisector to the given line.
Step 5: The line \(C D\) thus obtained is the perpendicular bisector of the line segment \(A B\). \(C D\) will bisect \(A B\) at point \(M\)

Parallel lines are lines, which never intersect or meet each other at any point. The steps of construction of parallel lines are listed below:

Step-1: Draw any line (Say \(PQ\)) of any length by using the ruler.

Step-2: Consider any point “\(R\)”, not on the given line.

Step-3: Draw a random line through the point \(R\) that intersects the given line by using the ruler.

Step-4: Draw an arc on the given line from the intersection point by using the compass of any radius and placing the tip of the compass at the intersection point of the given line and the line drawn through the point \(R\).

Step-5: With the same radius \((PX)\), draw an arc to the line drawn from the point \(R\) by using the compass by placing the tip of the compass at the given point \(R\).

Step-6: Again draw one more arc to intersect the previous arc from the intersection point of the arc and the line “\(S\)” by using the compass of the radius \((XY)\) and placing the tip of the compass at the intersection point \(S\).

Step-7: Now draw a line through the given point “\(R\)” and the intersection point formed by two arcs \(T\).

Here, the line \(RT\) gives the parallel line drawn to the given line \(PQ\).

Solved Examples: Construction of Parallel and Perpendicular Lines

Q.1. Construct the perpendicular line to the given line \(PQ\).
Ans: Steps of construction:
Step-1: Draw one line \(P Q\) of any length by using the ruler.
Step-2: Place any point (Say \(R\)) not on the line segment.
Step-3: By keeping the tip of the compass at the point and taking any radius, draw an arc that cuts the given line segment at \(A\) and \(B\).
Step-4: Draw circles taking the intersection points \(A\) and \(B\) as centres by using the compass with the radius equal to more than half of the length of the line.
Step-5: Draw the line through the intersection points of the two circles and the given point as shown below:

Here, the blue colour lines shown in the figure are perpendicular lines.

Q.2. Construct the line that intersects the other line at any point of the line with an angle of \(90^\circ \) by using the protractor and name the lines so formed.
Ans: We know that a line intersecting another line at an angle of \(90^\circ \) is perpendicular to the given line. Steps of construction for drawing such line are given below:1. Draw any line by using the ruler and locate any point on the line (Say \(P\)).

2. Now place the protractor at the point on the line such that the baseline of the protractor lies on the line, and the centre point of the protractor should be kept at the given point (Say \(P\)).

3. Now measure the \(90^\circ \) on the protractor mark the point (Say \(B\)) with the pencil.
4. Now join the points \(P\) and \(B\) by using a ruler.

Thus, the line \(PQ\) drawn is the required line that is perpendicular to the given line.

Q.3. Construct a parallel line through the point \(D\), to the given line \(AB\).
Ans: Steps of construction:
1. Consider the given line \(AB\) and the point \(D\).

2. Draw a random line through point \(D\) that intersects line \(AB\) by using the ruler.

3. Draw an arc by placing the tip of the compass at point \(A\) of any radius.

4. With the radius \((AX)\), draw an arc at the point \(D\), to the line drawn from the point by using the compass. Consider the intersection point as \(E\).

5. Again draw one more arc at the point \(E\), to intersect the previous arc by using the compass of the radius \(XY\).

6. Now draw a line through point \(D\) and the intersection point \(C\) formed by two arcs and extend it up to \(F\).

Here, line \(DF\) gives the parallel line drawn to the given line \(AB\) through point \(D\).

Q.4. Draw a perpendicular bisector to the line segment of length \(5 \,\text {cm}\).
Ans: Steps of construction:
1. Draw a line segment (say \(AB\)) of length \(5 \,\text {cm}\) using the ruler.

2. Draw arcs from point \(A\), above and below the line segment.
3. Again, draw arcs from point \(B\) to intersect the previous arcs at points \(C\) and \(D\).

4. Join the points \(C\) and \(D\). \(CD\) intersects at the point \(M\)

Thus, the line \(CD\) is the perpendicular bisector to the line segment \(AB=5 \,\text {cm}\) at the point \(M\).

Q.5. Observe the given figure and check which lines are parallel and which are perpendicular.

Ans: The lines \(a x+b y+c=0\) and \(a x+b y+d=0\) are non intersecting each other, and they have the same distance between them.
So, the given lines shown in the below figure are parallel lines.

The lines \(a x+b y+c=0\) and \(b x-a y+d=0\) in the graph are intersecting, and they are touching at the right angle.
So, the line shown in the below figure is perpendicular lines.

In this article, we have discussed the definitions of parallel lines, perpendicular lines. This article gives the construction of perpendicular lines and perpendicular bisector of the line. This article also studies the construction of the parallel lines with the necessary steps. We learned how to construct a perpendicular to a line through an external point, a point on the line; how to construct a perpendicular bisector of a line; how to draw parallel lines using a compass; how to draw parallel and perpendicular lines using a protractor and more. This article also gives the solved examples that help us to understand the constructions easily.

Learn All About Parallel Lines

Frequently Asked Questions (FAQs)

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

Q.1. How do you construct the parallel lines? Ans: (Refer to the image given below)1. Draw a line \((P Q)\) and locate the point \((R)\), not on it.2. Now draw the line from point \((R)\) to line \(P Q\).3. Draw an arc \((X Y)\) from point \(P\).4. Draw an arc from point \((R)\) with radius \(P X\) on the line \(P R\).5. Again draw another arc from point \(S\) of radius \(X Y\).6. Join the points \(R\) and \(T\), which give the parallel line to the given line.

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