A perpendicular bisector is a line segment that meets another line segment perpendicularly and divides it into two equal-sized halves. The Construction of the Perpendicular Bisector is done with the help of a ruler, a compass, and a pencil.

A perpendicular bisector bisects a line segment from the middle or through the mid-point. It makes a 90° angle on both sides of the bisected line segment. This article will learn the construction of a perpendicular bisector on a line segment.

Learn the Concepts of Perpendicular Bisectors

Perpendicular Lines

Perpendicular lines are lines that intersect at right angles, i.e., at \({90^ \circ }.\) In simple words, we can say that two lines are said to be perpendicular if they intersect such that the angles formed between them are at right angles. In the below-given figure, the lines \(l\) and \(m\) are perpendicular lines to each other.

Perpendicular to a Line 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 effortless to have a perpendicular to \(l\) through \(P.\)
Try this activity to get the perpendicular line.
We can fold the paper such that the lines received from both sides of the fold overlap each other. Now, let us take a tracing paper or 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 marked point \(P.\) Open out, and the crease is perpendicular to \(l.\)

Now let us learn to draw a perpendicular to any given line with the help of compasses and a ruler.

How to Draw the Perpendicular Bisector?

Drawing the Perpendicular on to a Given Line from a Given Point Outside the Line

Let \(XY\) be the given line and \(P\) the given point lying outside the line \(XY.\)

Steps of Construction:

1. Draw \(XY\) a line segment and mark a point \(P\) outside it.
2. With \(P\) as a centre and a convenient radius, draw an arc intersecting \(XY\) at \(A\) and \(B.\)
3. With \(A\) as the centre and a radius greater than \(\frac{1}{2}AB,\) draw an arc.
4. With \(B\) as a centre and the same radius, draw another arc, cutting the previous arc at \(Q.\)
5. Join \(PQ,\) meeting \(XY\) at \(L.\) Then, \(PL\) is the required perpendicular on \(XY.\)

Drawing the Perpendicular on to a Given Line Through a Given Point on the Given Line

Let \(AB\) be the given line and let \(M\) be a pint on the line \(AB.\)

Steps of Construction:

1. Taking \(M\) as the centre, draw two arcs of the same radii. Let these arcs cut \(AB\) at points \(P\) and \(Q.\)
2. Now taking \(P\) and \(Q\) as centres, draw arcs of equal radii intersecting at point \(X.\)
3. Join \(M\) and \(X.\)
\(MX\) is the required perpendicular to the line \(AB\) through point \(M\) on it.

Perpendicular Bisectors of a Line Segment

Now, as we are thorough with the concept of perpendicular lines and also learned various methods to construct perpendicular lines, let us now learn about the concept of perpendicular bisector.
In a plane, the perpendicular bisector of a segment is the line perpendicular to the segment at its midpoint. Line \(l\) is the perpendicular bisector of line segment \(AB.\)

Construction of a Perpendicular Bisector of a Segment

Using Ruler and Compasses, to Construct the Perpendicular Bisector

Let us draw a perpendicular bisector to line \(AB.\)

Steps of Construction:

1. Open the legs of the compasses to more than half the length of \(AB.\) With \(A\) as the centre, draw arc \(1.\)
2. With \(B\) as a centre and the same radius, draw arc \(2\) to cut the first arc. Name the points of intersection as \(P\) and \(Q.\)
3. Draw a line through \(P\) and \(Q\) by joining \(P,Q.\) This line bisects the given line segment \(AB\) and is called the bisector of \(AB.\)
Let \(PQ\) cut \(AB\) at \(M.\) Then \(M\) is called the middle point or simply the midpoint of \(AB.\) Thus, the line \(PQ\) is the perpendicular bisector or the right bisector of \(AB.\)

Draw the Perpendicular Bisector of a Given Line Segment

Let \(AB\) be the given line segment.

Steps of Construction:

1. With \(A\) and \(B\) as centres, draw arcs of equal radii on both sides of \(AB.\) The radii of these arcs must be more than half the length of \(AB.\)
2. Let these arcs cut each other at point \(C\) and point \(D.\)
3. Join \(CD,\) which cuts \(AB\) at \(M.\)
4. Then, \(AM = BM\) and \(\angle AMC = {90^ \circ }.\) Hence, the line segment \(CD\) is the perpendicular bisector of \(AB\) as it bisects \(AB\) at \(M\) and is perpendicular to \(AB.\)

Solved Examples

Q.1. Draw a line segment of \(6\,{\text{cm}}.\) Construct its perpendicular bisector.
Ans: The steps of construction are as follows:
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.2. Draw a perpendicular on a line \(l\) from the point \(A\) lying outside the line \(l.\)
Ans: We have to draw a perpendicular from the point \(A\) to the line \(l.\)
Open the compass a radius larger than the perpendicular distance from the point to the line, and scribe an arc intersecting the line at two points. Call the two points \(P\) and \(Q.\)
Scribe arcs from \(P\) and \(Q\) of the same radius on the other side of the line. Let the point where the arcs intersect be \(B.\) Join \(AB.\)
\(AB\) is perpendicular to the line \(l\) through the point \(A.\)

Q.3. Construction of the perpendicular bisector of a line segment is not always possible. Yes or No?
Ans: Perpendicular bisector can be drawn using a ruler and compass. We can draw a perpendicular bisector of any line segment. So, the construction of the perpendicular bisector of a line segment is always possible.

Q.4. Draw a line segment of \(10\,{\text{cm}}.\) Draw its perpendicular bisector.
Ans: Draw a line segment \(AB\) of \(10\,{\text{cm}}.\) with the help of a ruler.
Using a compass and taking a radius more than half of the length of the line segment, draw arcs above and below the line segment from \(AB\) and \(B.\)
Mark the points where the arcs coincide as \(C\) and \(D\) respectively. Join \(CD\)
\(CD\) is the perpendicular bisector of the given line segment.

The above figure shows the perpendicular bisector of a given line segment with a compass and a ruler.

Q.5. On a line segment of \(9.6\,{\text{cm}},\) a perpendicular bisector is drawn. What will be the measure of both the bisected parts?
Ans: We know that a perpendicular bisector may be defined as the line segment that intersects some other given line perpendicularly, and it also divides it into two congruent or equal parts. Thus the measure of each part will be equal to \(\frac{{9.6}}{2}~{\text{cm}} = 4.8~{\text{cm}}\)
Hence, the measure of each part will be equal to \(4.8~{\text{cm}}\)

Summary

In this article, we first learnt about the perpendicular lines and how to construct a perpendicular line on the given line segment. Moving forward, we learnt the definition of perpendicular bisector with the help of an example and activity. We then learnt the method to draw the perpendicular bisector with the help of a ruler and compasses. Lastly, we solved some examples to get a clear picture to construct a perpendicular bisector of a line segment.

Learn About Perpendicular Bisector of Triangle

FAQs

Q.1. What is a perpendicular bisector of a segment?
Ans: In a plane, the perpendicular bisector of a segment is the line perpendicular to the segment through its midpoint.

Q.2. 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 divide the line segment into two congruent or equal parts. A perpendicular bisector may be defined as the line segment that intersects some other given line perpendicularly, and it also divides it into two congruent or equal parts. So, while constructing a bisector of a line segment, we are also constructing the perpendicular bisector of the segment.

Q.3. Where does the perpendicular bisector of the line segment lie?
Ans: The perpendicular bisector of a line segment lies in the midpoint of the line segment and also cut that line segment at right angles, i.e., \({90^ \circ }.\)

Q.5. Which angle is made when we construct a perpendicular bisector for a line?
Ans: When we construct a perpendicular bisector, the bisected line makes an angle of \({90^ \circ }.\) with the line segment.

Q.5. Define perpendicular lines.
Ans: Two lines are said to be perpendicular if they intersect such that the angles formed between them are right angles.

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