Need Help?

Get in touch with us

bannerAd

Angle Bisectors of Triangles

Sep 12, 2022
link

Key Concepts

  • Angle Bisector Theorem
  • Concurrency
  • In-center

Introduction 

In this session, you will learn about angle bisector theorem, its converse, concurrency of angle bisectors of a triangle, and incenter. 

What is an angle bisector? 

What are segments? 

Angle Bisector Theorem 

Angle Bisector

A ray that divides an angle into two adjacent congruent angles is called the angle bisector. 

Distance from a Point to Line

The length of the perpendicular segment from the point to the line is called the distance from a point to the line. 

parallel

Angle Bisector Theorem

If a point is on the angle bisector, then it is equidistant from the two sides of the angle. 

If OC is the angle bisector of ∠AOB, then CA ⊥ OA and CB ⊥ OB, 

Then CA = CB  

Angle Bisector Theorem

Converse of Angle Bisector Theorem 

Theorem

If a point is in the interior of an angle and is equidistant from the two sides of the angle, then the point lies on the angle bisector. 

Converse of Angle Bisector Theorem  Theorem

If CA ⊥ OA and CB ⊥ OB, CA = CB, then C lies on angle bisector of ∠AOB  

parallel

Example 1: 

Find ∠DAC in the given figure if CD = BD = 10 units. 

Example 1: 

Solution: 

If CD ⊥ AC and DB ⊥ AB, DC = DB   

D lies on angle bisector of ∠CAB  

By the converse of angle bisector theorem 

∠DAC= ∠DAB 

= 52°

Example 2: 

In the figure ABCD, if D bisects ∠BAC, find CD. 

Example 2: 

Solution: 

D bisects ∠BAC, and  

D is equidistant from the sides of the angle 

So, CD = DB 

Since by using an angle bisector theorem. 

CD = DB 

3x + 5 = 2x + 10 

3x – 2x = 10 – 5  

x = 5 

CD = 3x + 5  

= 3*5 + 5 

= 15 + 5  

= 20 units 

Therefore, D lies on the bisector of ∠BAC, if CD = 20 units.  

Concurrency of Angle Bisectors of a Triangle 

Theorem

The angle bisectors of a triangle intersect at a common point that is equidistant from the sides of a triangle. 

If AP, BP, CP are angle bisectors of the triangle, then XP = YP = ZP 

Concurrency of Angle Bisectors of a Triangle Theorem

Example: 

In the figure, if P is the incenter, then find PY. 

Example: 

Solution: 

Here, we use the point of concurrency for angle bisectors of a triangle theorem. 

From that theorem, P is equidistant from the three sides of triangle ABC, so XP = YP = ZP 

We must find YP, for that we can find XP from ∆ XBP, 

BX = 5 units, BP = 13 units  

As, ∆ XBP is a right-angled triangle, we use Pythagorean theorem, 

BP2 = XP2 + BX2

132 = XP2 + 52

132−52 = XP2

169−25 = XP2

XP2 = 144

√XP = 144

XP = 12

Incenter 

The angle bisectors of a triangle intersect at a common point that is equidistant from the sides of the triangle, this point of concurrency is called the incenter of the triangle. 

Incenter always lies inside the triangle, the circle drawn with incenter as the center is inscribed in the triangle and it is called incircle. 

Angle Bisectors in Real Life  

Angle bisectors in real life are used in architecture, for making quilts and in preparing cakes. 

Angle bisectors are used in playing several games as well like rugby, football, billiards. 

Real Life Example  

In Snooker game, the position of the cue is relative to the ball and goal pocket forms congruent angles, as shown.  

Will the Snooker player have to move the cue farther to shoot toward the right goal pocket or the left goal socket? 

Real Life Example  

Solution: 

The congruent angles tell that the cue is placed on the angle bisector of the balls.  

By the angle bisector theorem, the snooker player’s cue is equidistant from both left goal pocket and the right goal pocket.  

So, the Snooker player must move the same distance to shoot another goal. 

Exercise

  • From the given figure, find the relation between AD and ∠BAC.
From the given figure, find the relation between AD and ∠BAC.
  • From the given figure, if BD ⊥ AB, DC ⊥ AC, then find the relation between BD and DC.
From the given figure, if BD ⊥ AB, DC ⊥ AC, then find the relation between BD and DC
  • From the given figure, find relation between P and ∠NOM.
From the given figure, find relation between P and ∠NOM
  • If P is equidistant from all the sides of the triangle ABC, a circle drawn with P as center touching all the sides, then the center of the circle is also called ______.
  • In a scalene triangle, the incenter lies ____.
  • Prove the converse of the angle bisector theorem.
  • Can you do an activity to show the location of the incenter?
  • Identify the theorem used to solve the problem.
Identify the theorem used to solve the problem.
  • Find ∠DAC in the given figure.
Find ∠DAC in the given figure
  • Find MP in the given figure.
Find MP in the given figure

Concept Map

Concept Map

What we have learned

  • Angle bisector theorem.
  • Converse of angle bisector theorem.
  • Concurrency.
  • Incentre

Comments:

Related topics

Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Read More >>
special right triangles_01

Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem?  Right Angle Triangles A triangle with a ninety-degree […]

Read More >>
simplify algebraic expressions

Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

Read More >>
solve right triangles

How to Solve Right Triangles?

In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles.  Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]

Read More >>

Other topics