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# Bisector|Definition & Meaning

## Definition

In geometry, a **bisector** is any shape or line that **cuts** a **line segment, an angle, or a curve** into **two equal parts**. This can be explained by **bisecting** a line segment using a bisector, where a line cuts a line segment **AB** into two equal pieces. It means that the line segment is bisected, that is, split into two equal parts. A similar case can be considered where an **angle** is bisected using a line segment.

Furthermore, a bisector can also be in the form of a **3-D plane.** This **plane **cuts another 3-D plane into two equal and **symmetrical **parts. There are multiple types of bisectors in geometry.

A bisector can cut a line segment at any angle to the line segment given that it divides the line into two equal parts. A bisector that is perpendicular to the line segment that is getting bisected is called a **perpendicular bisector. **

A line that **slices** an angle formed by** two adjacent lines** equally is known as an **angle bisector.** This signifies that the angle is formed by two intersecting lines or by three points **A, B, **and** C**, with the angle formed by two line segments, split in half by an angle bisector **DB**, where **D **is a point on the graph.

## Different Cases of Application of a Bisector and Their Construction

Let us consider we are given a line **AB** that is** 10 centimeters** long. If we use a bisector at any angle across this **line AB**, it will intersect the line at a midpoint M and divide the line segment into two lines namely line **AM** and **MB**. These two lines will be half of the original length which is 5 cm. Hence, a bisector is very helpful in dividing a line segment into equal parts.

In another case, let us say we have two adjacent line segments **AB **and **BC** making an angle of **80°**. This **∠ABC** can be cut into half as per our requirements by drawing an angle bisector. This can be done by **drawing two arcs** from the same distance on the line segment from the origin point **B** into the middle of the two line segments.

The **intersecting point** between these two arcs will decide the point **D** and the line segment** BD** will be our **angle bisector**. This can be easily done with the help of a compass.

Furthermore, a perpendicular bisector can also be easily constructed using a compass and a ruler. Firstly, you need to draw **two arcs of a specific radius** using a compass on **each side** of the line segment with the **center being the end of the line segment**. The same arcs are then drawn again from the other side of the line segment is the center of the arc.

The arcs** intersect at two points**, one on each side of the line segment. Then all you need is to draw a **straight line through the two points**. The resulting line is the perpendicular bisector of the corresponding line segment.

## Significance of a Bisector in Geometry and Advanced Real-Life Applications

A **bisector** is of utter importance in many **geometrical theorems** and is fundamental to many **concepts** in geometry. Most commonly, **triangular propert****ies** can be easily found by finding its **centroid, median, incenter,** and** circumcenter** and bisectors play an important part in finding these properties.

The section connecting a **vertex **to the **middle of the other side **is known as a triangle’s **median**. A median is hence a bisector of that opposing side. The **centroid of a triangle** is the place where all of the **medians connect**, and this is particularly significant since it establishes the triangle’s **center of mass**, given that the triangle is uniform.

The **circumcenter** is the center of the circle (also known as the** circumcircle**) that goes across the triangle’s three vertices. Furthermore, the circumcenter is also the intersecting point of all three perpendicular bisectors that **bisect each side** of the triangle.

Moreover, a circle enclosed inside the triangle has a center point that can be found by **intersecting all three of the angle bisectors** of each vertex of the triangle. This is called the **incenter** of the triangle.

The centroid, circumcenter, and incenter of the triangle are three of the **four common centers of a triangle** that are used to define the **property of a given triangular figure**. They all differ in the type of bisectors used, and their intersection points define that center.

In figure 5 above, the blue circle is the circumcircle that has a circumcentre at point F, which is made by the intersection of perpendicular bisectors. The green circle inside the triangle has a center at the incenter of the triangle at point D, made by the intersection of the angle bisectors of the triangle.

## An Example of Calculating a Bisector of a Line Segment

Given is a line **AB **of length **20 centimeters**. Divide this line into** four equal pieces** by drawing perpendicular bisectors across it.

### Solution

The** midpoint** **M** of the line **AB** can be calculated as follows:

\[ M = \frac{(20 + 0) }{2} \]

\[ \mathbf{M = 10 cm}\]

Thus, we will draw a perpendicular line at point **M** that will be our first bisector, and two line segments **AM **and **MB** are formed. We will find the midpoints of these two line segments** P** and **Q** to divide them further into **two more** equal parts.

\[ P = \frac{(10 + 0) }{2} \]

\[ \mathbf{P = 5 cm}\]

\[ Q = \frac{(20 + 10) }{2} \]

\[ \mathbf{Q = 15 cm}\]

Hence, we will now draw a perpendicular line across the points **P **and **Q** at the** 5 cm **and **15 cm** **mark **respectively. Thus we are given a line segment **AB** that is divided **equally into 4 parts.**

*All mathematical drawings and images were created with GeoGebra.*