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A perpendicular bisector is a straight line that is at a right angle with a given line or makes an angle of $90^{\circ}$ with another line and bisects (divides into two equal parts) it. Perpendicular bisectors are frequently used in geometry, especially in triangles, and have wide applications.

Let’s understand how to construct a perpendicular bisector with steps and examples.

## Properties of a Perpendicular Bisector

The following are the important properties of a 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 ends of the segment that they bisect.
- Can be only one in number for a given line segment.

## How to Construct a Perpendicular Bisector of a Segment?

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

The following steps are used to construct a perpendicular bisector of a line segment.

**Step 1:** Draw a line segment $\text{AB}$ of any suitable length.

**Step 2:** Take a compass, and with $\text{A}$ as the centre and with more than half of the line segment $\text{AB}$ as width, draw arcs above and below the line segment.

**Step 3:** Repeat the same step with $\text{B} as the centre.

**Step 4:** Label the points of intersection as $\text{P}$ and $\text{Q}$.

**Step 5:** Join the points $\text{P}$ and $\text{Q}$. The point at which the perpendicular bisector $\text{PQ}$ intersects the line segment $\text{AB}$ is its midpoint. Label it as $\text{O}$.

In the above figure $\text{PQ}$ is the perpendicular bisector of a line segment $\text{AB}$, where $\text{PQ}$ is perpendicular to $\text{AB}$ and $\text{PQ} \perp \text{AB}$.

## 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 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.

### Steps to Construct Perpendicular Bisector of a Triangle

The following steps are used to construct perpendicular bisectors of a triangle.

**Step 1:** Draw a triangle and label the vertices as $\text{A}$, $\text{B}$, and $\text{C}$.

**Step 2:** With $\text{B}$ as the centre and more than half of $\text{BC}$ as the radius, draw arcs above and below the line segment, $\text{BC}$. Repeat the same process without a change in radius with $\text{C}$ as the centre.

**Step 3:** Label the points of intersection of arcs as $\text{X}$ and $\text{Y}$ respectively and join them. This is the perpendicular bisector for one side of the triangle $\text{BC}$.

**Step 4:** Repeat the same process for sides $\text{AB}$ and $\text{AC}$. All three perpendicular bisectors make an angle of $90^{\circ}$ at the midpoint of each side.

**Step 5:** The perpendicular bisector of a triangle after construction is shown below. $\text{AD}$, $\text{BE}$, and $\text{CF}$ are the perpendicular bisectors of sides $\text{BC}$, $\text{AC}$, and $\text{AB}$ respectively.

**Note:** The point of intersection of three perpendicular bisectors of a triangle is called the circumcentre of a triangle. In the above figure, $\text{G}$ is the circumcentre of $\triangle \text{ABC}$.

## Practice Problems

- What is a perpendicular bisector?
- How many perpendicular bisectors can a triangle have?
- The point of intersection of all the perpendicular bisectors of a triangle is called ________.
- How will you draw a perpendicular bisector of a given line segment?

## FAQs

### What is the perpendicular bisector in Geometry?

A 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 $90^{\circ}$ with the line segment it bisects.

### How many perpendicular bisectors can be constructed for a line segment?

There can be only one perpendicular bisector constructed for a given line segment. This is because there can be only one midpoint for a line.

### Can a perpendicular bisector always be a median of a triangle?

No, a perpendicular bisector is not always a median of a triangle.

A perpendicular bisector of a triangle is a line segment drawn from a vertex perpendicular to the opposite side, while a median is a line segment drawn from a vertex and bisecting the opposite side.

A Perpendicular bisector can be a median of a triangle only in the case of an equilateral triangle.

### What is the difference between a perpendicular bisector and an 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 8 cm makes two line segments of 4 cm each, whereas, an angle bisector for a given angle of $30^{\circ}$ bisects the angle and makes two angles of $15^{\circ}$ each.

### What are the properties of a perpendicular bisector of a chord of a circle?

The following are the properties of a perpendicular bisector of a chord of a circle

a. Bisects the chord of a circle.

b. Makes an angle of $90^{\circ}$ with the chord.

c. Passes through the center of the circle.

## Conclusion

A perpendicular bisector is a straight line that is at a right angle with a given line or makes an angle of $90^{\circ}$ with another line and bisects (divides into two equal parts) it. We can construct a perpendicular bisector of a line segment using a ruler and a compass. For a triangle, three perpendicular bisectors can be constructed and the point of intersection of three perpendicular bisectors of a triangle is called the circumcentre of a triangle.

## Recommended Reading

- What is a Circle – Parts, Properties & Examples
- How to Construct a Perpendicular Line (With Steps & Examples)
- How to Construct Parallel Lines(With Steps & Examples)
- How To Construct a Line Segment(With Steps & Examples)
- What are Collinear Points in Geometry – Definition, Properties & Examples
- What is a Transversal Line in Geometry – Definition, Properties & Examples
- What are Parallel Lines in Geometry – Definition, Properties & Examples
- What is Concurrent lines in Geometry – Definition, Conditions & Examples
- What is Half Line in Geometry – Definition, Properties & Examples
- What is a Perpendicular Line in Geometry – Definition, Properties & Examples
- Difference Between Axiom, Postulate and Theorem
- Lines in Geometry(Definition, Types & Examples)
- What Are 2D Shapes – Names, Definitions & Properties
- 3D Shapes – Definition, Properties & Types