Construction of Perpendiculars: Definitions, Constructions, Examples

Construction of Perpendiculars: When two lines intersect each other to create right angles, these lines are said to be perpendicular to each other. It means that if you have two lines perpendicular to each other, the angles between two lines are right angles. By using a ruler and a compass, you can easily construct a line perpendicular to the other that passes through a point that lies on that line. But, this may not always be accurate. We have other accurate and precise ways for the construction of perpendiculars. Here, we will learn about the construction of perpendicular lines step by step in detail.

Perpendicular Lines

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

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

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

Learn About Perpendicular Bisector Here

Perpendicular to a Line Segment Through a Point on it

We were given a line \(l\) drawn on a paper sheet and a point \(P\) lying on the line. It is easy to get a perpendicular to \(l\) through \(P\).

Let us try this activity to get the perpendicular lines.

We can fold a paper such that the lines received from both sides of the fold intersect or overlap each other.

Now, let us take a transparent paper and draw any line \(l\) on it. Let us mark a point \(P\) anywhere on \(l\).

Fold the sheet such that \(l\) is reflected on itself. Adjust the fold so that the crease passes through the point \(P\). Open out, and we can see that the crease is perpendicular to \(l\).

Construction Tools

The geometrical figures are constructed by using some specified rules and steps. The different tools used to construct the geometrical shapes are given below:

Ruler or Scale
In construction, we use it to draw the line segments. Also, we use it to measure the length of the sides of the geometrical plane figures and the line segments during constructions.

Protractor
Protractor is the tool that we use to draw or measure the angles of any measure.

Compass
It is the construction tool that is used to draw the arcs.

Construction of Perpendicular Line Through an External Point

The steps of constructing the perpendicular lines are given below:

Steps of construction:

1. Draw a line segment \((P Q)\) of any length by using the scale or ruler.

2. Place any point \((R)\), outside the line segment \(PQ\).

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 \((X, Y)\).

4. Draw circles at the intersection points \((X, Y)\) of arc and line segment using the compass with a radius that is equal to more than half of the length of the line segment \((XY)\) between the intersecting points.

5. Construct the line segment through the intersection points of the two circles \((A, B)\).

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

Construction of Perpendicular From a Point on it

Steps of Construction

1. Let us draw any line \(P Q\) and point \(M\) that we can assume the midpoint on it.
2. Draw arcs on any side of the point \(M\) on the line \(P Q\) using a compass of any radius and mark the intersection points as \(A\) and \(B\).
3. By considering the radius equal to more than half of the line segment \(A B\), construct arcs from point \(A\) on top of the line.
4. With the equal radius, draw an arc from point \(B\) on the bottom of the line.
5. Mark the intersection points of arc as \(C\). Now join the points \(C\) and \(M\).

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

Solved Examples – Construction of Perpendiculars

Q.1. Construct the perpendicular line to the given line \(PQ\).
Ans: Steps of construction:
Step-1: Draw a line \(P Q\) of any arbitrary length by using the ruler.
Step-2: Place any arbitrary 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 intersects the given line segment at \((X, Y)\).
Step-4: Draw two circles at the intersection points \((X, Y)\) of arc and line segment using the compass with a radius that is equal to more than half of the length of the line segment \((X Y)\) between the intersecting points.
Step-5: Draw the line segment through the intersection points of the two circles and the given point as shown below:

Here, the blue colour line segments shown in the figure are perpendicular to each other.

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 segment intersecting another line segment at an angle of \(90^{\circ}\) is perpendicular to the given line segment. Steps of construction for drawing such line are provided below:
1. Draw an arbitrary line segment using the ruler and locate any arbitrary point on the line (Say \(P\)).
2. Now fix the protractor at the point on the line segment such that the baseline of the protractor lies on the line segment, and the centre point of the protractor should be kept at the point \(P\).

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. Draw a line segment of \(6 \,\text {cm}\). Construct its perpendicular bisector.
Ans: The steps of construction are as provided below:
1. Draw a line segment of \(6 \,\text {cm}\).

2. With \(A\) as the centre, using compasses, draw a circle; the radius of your circle should be more than half the length of \(AB\).

3. With the same radius and with \(B\) as the centre, draw another circle using the compasses. Let it cut the previous circle at \(C\) and \(D\).
4. Join \(CD\). It cuts \(AB\) at \(O\).

Use the divider to verify that \(O\) is the midpoint of \(AB\).

Thus, \(CD\) is the perpendicular bisector of \(AB\).

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. Construction of the perpendicular bisector of a line segment is always possible. Yes or No?
Ans: Yes, a perpendicular bisector can be constructed using a ruler and compass. We can construct a perpendicular bisector of any arbitrary line segment. Thus, the construction of the perpendicular bisector of a line segment is always possible.

Summary

This article explained the definition of perpendicular lines. This article gives the construction of perpendicular lines to a line segment from an external point, perpendicular to a line segment from a point on it. This article provides some introduction about construction tools, the steps of construction of the perpendiculars using the tools. This article provides the solved examples that help us to understand the construction of perpendiculars easily.

Frequently Asked Questions (FAQs) – Construction of Parallel and Perpendicular Lines

Q.1. Define perpendicular lines.
Ans: Two lines are said to be perpendicular if they intersect such that the angles formed between them are right angles.

Q.2. How do you construct the perpendicular line?
Ans: 1. Construct a line and locate any arbitrary point, not on the line.
2. Construct a circle arc of any radius from the point intersecting the given line segment at two points.
3. From two points of the line segment, constructs two circles.
4. Join the point with the intersection points of two circles.

Q.3. When constructing the bisector of a line segment, are you also constructing the perpendicular bisector of the segment?
Ans: While constructing a bisector of a line, we split up the line segment into two congruent or the same parts. A perpendicular bisector may be described as the line segment that intersects a line perpendicularly and divides it into two congruent parts. Thus, constructing a bisector of a line segment, we are also constructing the perpendicular bisector of the line segment.

Q.4. How do you determine the perpendicular lines?
Ans: If the angle between the lines is \(90^{\circ}\) or a right angle, then they are perpendicular lines.

Q.5. What are tools used to construct the perpendicular lines?
Ans: The tools that we use for constructions are ruler, compass, protractor, etc.

Learn About Parallel And Perpendicular Lines Here

We hope this detailed article on the Construction of Perpendiculars 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.