Bisector: Perpendicular Bisector, Angle Bisector, Line Segment, Construction and Examples

In general, a bisector divides something into two equal parts. In geometry a bisector is a line that divides a line or an angle into congruent or equal parts. It extends all the way to the segment’s bisectors and angle bisectors. The line segment bisector is a straight line that travels through the midway of a line section, whereas the angle bisector is a straight line that runs through the apex of an angle.

Also read: Isosceles Triangle Theorems

Key Terms: Line Bisector, Angle Bisector, Perpendicular, Triangle, Equidistant, Sides, Angle, Vertex

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What is a Bisector?

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A bisector is a straight line that breaks an angle or a line into two equal parts. A segment's bisector always passes through the sections halfway. Basically, the bisectors are classified into two types based on the geometrical shape that it bisects.

• Line Segment Bisector (Perpendicular Bisector theorem)
• Angle Bisector (Triangle Bisector Theorem)

Read more: Angle Formula

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Line Segment Bisector

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A line segment bisector breaks a line segment into two halves that are equal of length. It passes the line segment through its midpoint. Here the red line bisects the blue line segment.

​Line Segment Bisector

• Example for Line segment Bisector 

In the below given picture the line segment in the upper section of the image is approximately 10 inches long and the bottom section of the image shows what exactly happens how a line segment bisector breaks this line segment.

Example for Line segment Bisector 

As a result, the line segment which 10 inches long. It is split into two pieces, each 5 inches long, after being split by a line segment bisector. When the two 5-inch pieces are added together, the result is 5 inches + 5 inches = 10 inches.

Read more: Line Segment

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Perpendicular Bisector

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A perpendicular bisector is a line segment or a line that separates a given line segment into two equal-sized parts. The term 'bisect' is used to represent dividing evenly. Perpendicular bisectors contact the line segment they bisect on both sides and produce four 90° angles. A perpendicular line segment or a line is one that forms a 90° angle with another line segment or a line.

Perpendicular bisector

Properties of Perpendicular Bisector

• The important properties of a perpendicular bisector are mentioned below.
• Creates two congruent segments from a line segment or a line.
• Divides a triangle's sides into congruent sections.
• They form a 90-degree angle with the line being bisected.
• They cross the line segment exactly in the middle.
• The circumcentre of a triangle is the place where the perpendicular bisectors intersect.
• They meet inside a triangle in an acute triangle, outside the triangle in an obtuse triangle, and at the hypotenuse in right triangles.

Read more: Vertex

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Perpendicular Bisector Theorem

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The perpendicular bisector theorem states that “Any point that lies on the perpendicular bisector is equidistant from both the edges of the line segment on which it is drawn”.

Perpendicular Bisector Theorem

In the Above Figure: MQ = NQ, MR = NR, MS = NS, MT = NT

• Proof of Perpendicular Bisector Theorem

Draw the diagram where the point C is considered as an arbitrary point on AB's perpendicular bisector which intersects AB at the point D:

Triangles for proving perpendicular bisector

Comparing ΔACD and ΔBCD

• AD=DB //Given, CD is a bisector
• ∠ADC≅ ∠BDC //Side-Angle-Side postulate
• CD=CD //Common side
• ∠ADC =∠BDC = 90° //Given, CD is perpendicular to AB
• CA=CB //corresponding sides of congruent triangles

 Hence proved the Perpendicular Bisector Theorem.

• Perpendicular Bisector Equation

The General equation of Perpendicular Bisector is represented by:

\(y = (y_1 + y_2) /2 = \frac{x_2 - x_1}{y_2 - y_1}(x - (x_1 + x_2)/2)\)

Read more: Introduction to Constructions

• Converse of Perpendicular Bisector Theorem

The converse of the perpendicular bisector theorem states that “A point is on the perpendicular bisector of a line segment if it is equidistant from both endpoints of the line segment in the same plane”.

• Steps to Construct a Perpendicular Bisector

The steps involved in constructing a perpendicular bisector is mentioned below:

Step 1: Firstly, draw the line segment

Step 2: At one end of the line segment, place the compass.

Step 3: Set the compass to a length that is slightly longer than half the length of the line segment. Arcs should be drawn above and below the line.

Step 4: Draw arcs from the other end of the line, keeping the compass width the same.

Step 5: Draw the line segment with the ruler where the arcs cross.

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Angle Bisector

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A line that divides an angle into two sections of equal size. The ∠LKJ is bisected by the red line KM originating from the point K as shown in the diagram below. The bisector is the red line.

Angle Bisector

In the figure, the KM bisects the angle ∠JKL.

The angles ∠JKM and ∠LKM are congruent.

So, ∠JKM = ∠LKM.

• Example for Angle Bisector

 Example for Angle Bisector

In the diagram BD is an angle bisector. Determine the value of ABC.

Due to the fact that BD is an angle bisector,

∠CBD = ∠ABD

Therefore, ∠CBD = 65°

The Angle Addition Postulate states that

∠ABC = ∠ABD + ∠CBD 

Substitute.

∠ABC = 65° + 65° = 130°

• Angle Bisector Theorem

Angle Bisector Theorem states that "An angle bisector of a triangle divides the opposing side into two segments that are proportional to the other two sides of the triangle".

Angle Bisector Theorem

• Properties of Angle Bisector

1. The important properties of an angle bisector are mentioned below.
2. An angle bisector is a tool that divides a given angle into two equal halves.
3. Any point on an angle's bisector is equidistant from the angle's sides or arms.
4. It divides the opposite side of a triangle into the ratio of the other two sides measures.

Read more: Difference between Sequence and Series

• Angle Bisector of a Triangle

One angle bisector is a straight line that divides an angle into two congruent or equal angles in a triangle. Every triangle can have three angle bisectors, one for each vertex. The incentre of a triangle is the place where these three angle bisectors meet. The distance between the incentre and all of a triangle's vertices is the same. Take a look at the graphic below, which depicts a triangle's angle bisector. The angle bisectors of BAC, ACB, and ABC, respectively, are AG, CE, and BD. The point F is the incentre, or point of intersection, of all three bisectors, and it is located at an equal distance from each vertex.

Angle Bisector of a Triangle

• Steps to Construct an Angle Bisector

The steps involved in constructing an Angle Bisector is mentioned below:

Steps to Construct an Angle Bisector

Step 1: Draw any angle, for an instance take ∠ABC.

Step 2: Taking the point B as the centre and any appropriate radius, draw an arc to intersect the rays BA and BC at, say, E and D respectively.

Step 3: Now, taking the points D and E as centres and with the same radius as taken in the previous step, draw two arcs to intersect each other at the point F.

Step 4: Join the points B to F and extend it as a ray. This ray BF is the required angle bisector of ∠ABC.

Steps to Construct an Angle Bisector

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Things to Remember

• A line, segment, or ray that runs perpendicular to a segment and passes through its midway
• Two-line segments, rays, lines, or any combination of those that meet at right angles are perpendicular.
• Bisector is a term used to describe an object (a line, a ray, or a line segment) that divides another object (an angle or a line segment) into two equal pieces. 
• A line, ray, or line segment that bisects an angle or line segment at a right angle.
• Any point within the angle bisector is equidistant from the angle's sides.
• The perpendicular bisector's points are equidistant from both ends of the segment they bisect.

Read more: First Order Differential Equation

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