Tuesday, 1 August 2017

Triangles VI: What is special about Incentre of a Triangle?

In this post, we will discuss Incentre of a triangle which will help you solve geometric questions asked in SSC Exams.

Incentre of a triangle

There are a total of 4 centres of a triangle out of which incentre is the most important centre. One should know all the properties of incentre of a triangle to solve geometric questions in SSC Exams. In our previous posts, we have discussed centroid of a triangle, circumcentre of a triangle, orthocentre of a triangle. In this post, we will discuss Incentre of a triangle which will help you save time in SSC Exams.

What is an Incentre of a Triangle?

Incentre of a triangle is the point of intersection of the angle bisectors of the interior or exterior angles of the triangle. 

Angle Bisector:

An angle bisector is a line that bisects the angles into two equal parts.
There are two types of angle bisectors. 

1. Interior angle bisector
2. Exterior angle bisector

There are three angles in a triangle, hence there can only be three angle bisectors that can be drawn inside the triangle. The point of intersection of three angle bisectors is called as the incentre of a triangle.

incentre of a triangle

Here, I is the incentre of traingle ABC

Interior Angle Bisector 

Properties of Incentre 1:

The angle bisector divides the opposite side of the triangle into a particular ratio. The ratio in which the sides are divided is the ratio of the sides that contains the angle. 

incentre of a triangle

Properties of Incentre 2:

The incentre 'I' is equidistance from all the three sides. i.e. if a perpendicular is drawn from the centre 'I' to the sides, they will be equal in length. 

IE = ID = IF

incentre of a triangle

Properties of Incentre 3:

Incentre is the centre of the circle which is inscribed completely inside the triangle. Inradius is the radius from the incentre to the perpendicular drawn on the sides of a triangle.
Incentre is denoted by the letter 'r.'

r = IF = ID = IE

incentre of a triangle

Properties of Incentre 4:

The angle made by any side with the incentre is 90° + half the opposite angle. 

In a triangle ABC,
∠BIC = 90° + 1/2 ∠A 
∠AIC = 90° + 1/2 ∠B
∠AIB = 90° + 1/2 ∠C

incentre of a triangle

Properties of Incentre 5:

For equilateral triangle, r = a/ 2√3 units.

incentre of a triangle

Exterior Angle Bisector 

Properties of Incentre 1:

The Exterior angle is equal to the sum of the opposite two angles. 

In  a triangle ABC, 
Equation (1) All the angles in a triangle equal to 180°
∠A + ∠B + ∠C = 180°
⇒  ∠A + ∠B = 180° - ∠C

Equation (2) The two Supplementary angles equal to 180°
∠C + ∠C= 180°
⇒  C= 180° - ∠C

From Equations (1) and (2), we get; 

∠A + ∠B  = C

incentre of a triangle

Properties of Incentre 2:

The point of intersection of two exterior angle bisectors is represented by the letter "P."
The angle made by the exterior angle bisectors and the point of intersection is 90° - half the opposite angle.

In the triangle ABC 

∠BPC = 90° - ∠A/2

Hope these Properties of incenter of a triangle have helped you understand the details of incenter of a triangle very well. Do write in the comment section below on how this blog has helped you solve geometric questions asked on Incenter of a triangle in SSC Exams.

To practice questions on incentre of a triangle, download free practice questions.

incentre of a triangle
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