What is a perpendicular bisector of a segment?

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Table of Contents Show

What is a Perpendicular Bisector?
Properties of a Perpendicular Bisector
How to Construct a Perpendicular Bisector?
Perpendicular Bisector of Triangles
Related Articles
Solved Examples
Practice Questions
What is Perpendicular Bisector?
Perpendicular Bisector Definition
How to Construct Perpendicular Bisector?
Steps for Constructing Perpendicular Bisector
Perpendicular Bisector of a Triangle
Perpendicular Bisector Properties
FAQs on Perpendicular Bisector
How Do You Construct Perpendicular Bisector With a Straight Edge and a Compass?
Can a Perpendicular Bisector always be a Median of a Triangle?
What are the Properties of Perpendicular Bisector?
What is the Perpendicular Bisector Theorem?
What is the Difference Between Perpendicular Bisector and Angle Bisector?
What is the Perpendicular Bisector of a Triangle?
What are the Properties of Perpendicular Bisector of a Chord?
How Many Perpendicular Bisector Can Be Constructed For a Line?

Perpendicular Bisector is a line or a segment perpendicular to a segment that passes through the midpoint of the segment.

In the figure, A B ↔ is a perpendicular bisector of C D ¯ .

Any point on the perpendicular bisector is equidistant from the endpoints of the line segment.

A perpendicular bisector can be defined as a line that intersects another line segment perpendicularly and divides it into two parts of equal measurement. We can draw a perpendicular bisector using a rule, a compass and a pencil.

Two lines are said to be perpendicular to each other when they intersect each other at 90 degrees or at right angles. And, a bisector is a line that divides a line into two equal halves. Thus, a perpendicular bisector of a line segment AB implies that it intersects AB at 90 degrees and cuts it into two equal halves.

What is a Perpendicular Bisector?

A perpendicular bisector is a line that bisects another line segment at a right angle, through the intersection point. Thus, we can say, a perpendicular bisector always divides a line segment through its midpoint. The term bisect itself means dividing equally or uniformly.

Properties of a Perpendicular Bisector

It divides AB into two equal halves or bisects it.
It makes right angles with (or is perpendicular to) AB.
Every point in the perpendicular bisector is equidistant from point A and B.

While working with practical geometry, you will often find the application of perpendicular bisectors; say when you are asked to draw an isosceles triangle, or when you have to determine the centre of a circle, etc. Below are the steps to construct a perpendicular bisector of a line using a compass and a ruler.

How to Construct a Perpendicular Bisector?

You will require a ruler and compasses. The steps for the construction of a perpendicular bisector of a line segment are:

Step 1: Draw a line segment PQ.
Step 2: Adjust the compass with a length of a little more than half of the length of PQ.
Step 3: Place the compass pointer at point P and draw arcs above and below the line.

Step 4: Keeping the same length in the compass, place the compass pointer at point Q. Similarly, draw two arcs above and below the line keeping the compass pointer at Q.

Step 5: Mark the points where the opposite arcs cross as X and Y.

Step 6: Using a ruler, draw a line passing across X and Y.

The perpendicular bisector bisects PQ at a point J, that is, the length PJ is equal to JQ. And the angle between the two lines is 90 degrees.

Perpendicular Bisector of Triangles

The perpendicular bisector of a triangle is the line segment that is drawn from a vertex to the opposite side bisecting the side at a right angle. The perpendicular of a triangle is perpendicular to the sides drawn from the opposite vertices and divides the sides into two equal parts. The point at which all the three perpendicular bisectors of a triangle meets is called the circumcenter of a triangle.

Solved Examples

Q.1: If a line segment is of length equal to 8cm and a perpendicular bisector is drawn to it. What is the measure of each part of the line segment?

Solution: By the definition, we know, a perpendicular bisector intersects a line segment at the right angle and divides it into two equal parts.

Hence, the measure of each part of the line segment (8 cm), which is bisected by a perpendicular is 4 cm.

Q.2: Can we draw a perpendicular bisector, if the length of the radius of a circle is known?

Solution: Yes, we can draw the perpendicular bisector if we know the radius of the circle. Since the diameter of a circle is equal to twice of radius. Hence. we can draw a perpendicular bisector to the diameter of circle following the same steps.

Practice Questions

Draw a perpendicular bisector to a line segment equal to 12 cm.
Construct a perpendicular bisector of a line segment equal to 6.6 cm.

A perpendicular bisector can be defined as a line segment which bisects another line segment at 90 degrees. In other words, a perpendicular bisector intersects another line segment at 90° and divides it into two equal parts.

Yes, a perpendicular bisector can be a median of a triangle. A median is defined as a line segment from a vertex of a triangle to the midpoint of the side opposite that vertex. So, if the median joins the opposite side at 90 degrees, it will be the perpendicular bisector of that side. For example, for an equilateral triangle, the medians are always perpendicular bisectors.

The point at which the perpendicular bisectors of a triangle meet are known as the circumcenter of the triangle and it is equidistant from all the vertices.

Perpendicular bisector theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the segment’s endpoints.

Perpendicular bisector is a line that divides a given line segment exactly into two halves forming 90 degrees angle at the intersection point. Perpendicular bisector passes through the midpoint of a line segment. It can be constructed using a ruler and a compass. It makes 90° on both sides of the line segment that is being bisected.

In this article, we will discuss the concept of the perpendicular bisector and learn how to construct it. We will also explore the perpendicular bisector of a triangle and its properties. We shall solve a few examples in the end for a better understanding of the concept.

What is Perpendicular Bisector?

A perpendicular bisector is a line that bisects a line segment in two equal parts and makes an angle of 90 degrees at the point of intersection. In other words, we can say that a perpendicular bisector divides a line segment at its midpoint making an angle of 90 degrees. Let us go through the formal definition of it in the next section to understand its meaning in a better way.

Perpendicular Bisector Definition

A perpendicular bisector is defined as a line or a line segment that divides a given line segment into two parts of equal measurement. 'Bisect' is the term used to describe dividing equally. Perpendicular bisectors intersect the line segment that they bisect and make four angles of 90° each on both sides. Perpendicular means a line or a line segment making an angle of 90° with another line or line segment. In the figure shown below, the perpendicular bisector bisects the line segment AB into two equal halves.

How to Construct Perpendicular Bisector?

Perpendicular bisector on a line segment can be constructed easily using a ruler and a compass. The constructed perpendicular bisector divides the given line segment into two equal parts exactly at its midpoint and makes two congruent line segments.

Steps for Constructing Perpendicular Bisector

Follow the steps below to construct a perpendicular bisector of a line segment.

Step 1: Draw a line segment XY of any suitable length.
Step 2: Take a compass, and with X as the center and with more than half of the line segment XY as width, draw arcs above and below the line segment.
Step 3: Repeat the same step with Y as the center.
Step 4: Label the points of intersection as 'P' and 'Q'.
Step 5: Join the points 'P' and 'Q'. The point at which the perpendicular bisector PQ intersects the line segment XY is its midpoint. Label it as 'O'.

Perpendicular Bisector of a Triangle

The perpendicular bisector of a triangle is considered to be a line segment that bisects the sides of a triangle and is perpendicular to the sides. It is not necessary that they should pass through the vertex of a triangle but passes through the midpoint of the sides. The perpendicular bisector of the sides of the triangle is perpendicular at the midpoint of the sides of the triangle. The point at which all the three perpendicular bisectors meet is called the circumcenter of the triangle. There can be three perpendicular bisectors for a triangle (one for each side). The steps of construction of a perpendicular bisector for a triangle are shown below.

Draw a triangle and label the vertices as A, B, and C.
With B as the center and more than half of BC as radius, draw arcs above and below the line segment, BC. Repeat the same process without a change in radius with C as the center.
Label the points of intersection of arcs as X and Y respectively and join them. This is the perpendicular bisector for one side of the triangle BC.
Repeat the same process for sides AB and AC. All the three perpendicular bisectors make an angle of 90“ at the midpoint of each side.

The perpendicular bisector of a triangle after construction is shown below.XY, HG, and PQ are the perpendicular bisectors of sides BC, AC, and AB respectively.

Perpendicular Bisector Properties

Perpendicular bisectors can bisect a line segment or a line or the sides of a triangle. The important properties of a perpendicular bisector are listed below.

Perpendicular bisector,

Divides a line segment or a line into two congruent segments.
Divides the sides of a triangle into congruent parts.
They make an angle of 90° with the line that is being bisected.
They intersect the line segment exactly at its midpoint.
The point of intersection of the perpendicular bisectors in a triangle is called its circumcenter.
In an acute triangle, they meet inside a triangle, in an obtuse triangle they meet outside the triangle, and in right triangles, they meet at the hypotenuse.
Any point on the perpendicular bisector is equidistant from both the ends of the segment that they bisect.
Can be only one in number for a given line segment.

Important Notes on Perpendicular Bisector

A perpendicular bisector is a line that divides a given line segment exactly into two halves forming 90 degrees angle at the intersection point.
The perpendicular bisector of a triangle is considered to be a line segment that bisects the sides of a triangle and is perpendicular to the sides.
Perpendicular bisector on a line segment can be constructed easily using a ruler and a compass.

☛ Related Articles

Given below is the list of topics that are closely connected to the perpendicular bisector. These topics will also give you a glimpse of how such concepts are covered in Cuemath.

Perpendicular Bisector Theorem
Perpendicular Bisector of a Chord
Geometry

Example 1: Find at which point a perpendicular bisector bisects a line segment of length 10 units.

Solution:

A perpendicular bisector is a line that bisects a given line segment into two congruent line segments exactly at its midpoint. It is given that the line segment is of the length 10 units. So, the perpendicular bisector bisects the line segment exactly at 5 units and the line segment of 10 units is divided into two line segments of 5 units each.
Example 2: Can you find if the points of intersection of all the perpendicular bisectors for an obtuse triangle PQR with measurements as follows: PQ = 5 units, QR = 8 units, and PR = 9 units lies outside or inside the triangle?

Solution: The perpendicular bisector of any triangle bisects the sides at its midpoint. In a triangle, there are three perpendicular bisectors that can be drawn from each side. To find the perpendicular bisector of a triangle with the given sides, follow the steps given below.
- Draw a line segment PQ of length 5 units.
- With P as the center and more than half of PQ as radius, draw arcs above and below the line segment PQ. Repeat the same process with Q as the center. Join the points of intersection of these arcs.
- Repeat the same process and draw perpendicular bisectors for the sides QR and PR.
The construction of the perpendicular bisectors of the obtuse triangle is shown below.

We can clearly see that all the perpendicular bisectors intersect at a point outside the triangle.
Example 3: Draw a perpendicular bisector to the diameter of a circle whose radius is 4 units.

Solution:

Given, the radius of the circle = 4 units. Diameter = 2 × radius. So, diameter = 2 × 4 = 8 units. The steps to construct a perpendicular bisector on a diameter of a circle are as follows.
- Draw a line segment XY of length 8 units.
- Using a compass, with X as the center and the radius as more than 4 units, draw arcs above and below the line segment.
- Repeat the same process with Y as the center. Label the points of intersection of the arcs as P and Q.
- Join the points P and Q. This line is the perpendicular bisector for the diameter of the circle. Label the point of intersection of the perpendicular bisector with the diameter as O.
The construction discussed in the above steps is shown in the figure below.

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FAQs on Perpendicular Bisector

Perpendicular Bisector is a line segment that bisects a straight line segment into two congruent or equal segments. They divide the line segment exactly at its midpoint. Perpendicular bisector makes 90° with the line segment it bisects.

How Do You Construct Perpendicular Bisector With a Straight Edge and a Compass?

Perpendicular bisector is constructed using a straight ruler and a compass using the following steps:

Draw a line segment AB of any length with a straight edge or a ruler.
Using a compass, with A as the center and more than half of AB as radius draw arcs above and below the line segment AB.
Repeat the above step with B as the center.
Mark the points of intersection as P and Q.
Join the points P and Q with a straight edge.
The line connecting the points P and Q is the perpendicular bisector to the given line segment which makes 90° with it.

Can a Perpendicular Bisector always be a Median of a Triangle?

Perpendicular bisector can be a median of a triangle only in the case of an equilateral triangle. Median is a line segment joining the vertex of one side of the triangle to the midpoint of its opposite side. If the median line segment intersects the opposite side at exactly 90° then we can say that the median is a perpendicular bisector. Therefore, the median of a triangle can be a perpendicular bisector only if it makes 90 degrees with the side opposite to it.

What are the Properties of Perpendicular Bisector?

Few properties of the perpendicular bisector are listed below:

Perpendicular bisector divides a line segment into congruent segments.
They intersect a line segment or the side of a triangle exactly at its midpoint.
Divides the sides of the triangle into two equal parts.
Can be only one in number for a given line segment.
Any point on the perpendicular bisector is equidistant from both the ends of the segment that they bisect.
They make 90 degrees at the midpoint where it touches the line segment it bisects.

What is the Perpendicular Bisector Theorem?

Perpendicular bisector theorem states that any point on the perpendicular bisector is always equidistant to both the ends of the line segment to which it is perpendicular.

What is the Difference Between Perpendicular Bisector and Angle Bisector?

Perpendicular bisector divides a line segment into two equal halves, whereas, angle bisector divides a given angle into two congruent angles. For example, a perpendicular bisector to a line segment of measure 10 units makes two line segments of 5 units each, whereas, an angle bisector for a given angle of 60 degrees bisects the angle and makes two angles of 30 degrees each.

What is the Perpendicular Bisector of a Triangle?

Lines that divide the sides of the triangle into two congruent segments are called perpendicular bisectors of a triangle. There can be three perpendicular bisectors for a triangle. They all meet at a point called circumcenter. It is not necessary that they pass through the vertex of a triangle to its opposite side's midpoint. Sometimes the perpendicular bisectors originate from a point that is away from the vertex and intersects the opposite side exactly at its midpoint. In an equilateral triangle, the medians of the triangle are perpendicular bisectors as they make 90 degrees with their opposite sides.

What are the Properties of Perpendicular Bisector of a Chord?

Perpendicular bisector to a chord:

Bisects the chord of a circle.
Makes 90 degrees with the chord.
Passes through the center of the circle.

How Many Perpendicular Bisector Can Be Constructed For a Line?

There can be only one perpendicular bisector constructed for a line. This is because there can be only one midpoint for a line. This fact is true because the perpendicular bisectors pass through the midpoint of a line or a line segment.