Geometry Concepts: Triangles, Its Types and Properties

This is the third post of geometry concept series in which we will learn about the triangles, their types, their properties and important theorems and results. So what is a triangle?

It is a three-sided closed plane figure which is formed by joining three non-collinear points. It is denoted by the symbol △.


In the given △ABC
  • A, B and C are the three vertices of this △ABC.
  • ∠A, ∠B and ∠C are interior angles of this triangle. 
  • ∠FCB, ∠CBE, ∠EBD, ∠DBA, ∠BAI, ∠IAH, ∠HAC, ∠ACG and ∠GCF.
  • AB, BC and CA are the three sides of this triangle.
  • Sum of the interior angles of a triangle is 180° i.e, ∠A + ∠B + ∠C = 180° 
  • Sum of three ordered exterior angles is 360°. That is ∠FCB + ∠DBA + ∠HAC = 360°. And similarly ∠ACG + ∠IAB + ∠CBE = 360°
  • Sum of all the sides of this triangle is equal to its perimeter i.e, AB + BC + CA = Perimeter.
  • Semiperimeter of a triangle is half of its perimeter. 
Types of Triangles

All the triangles can be classified under two heads
  • Interior angles
  • Length of sides 
Triangles according to interior angles
  • Acute angle triangle: Each of the angles of a triangle is acute angle (i.e. less than 90°)
  • Right angled triangle: One of the angles of a triangle is 90° and the remaining two angles are complementary to each other that means each of the remaining angles will be lees than 90°. 
  • Obtuse angle triangle: One of the angle is obtuse angle (i.e. more than 90°). 
Triangles according to length of sides
  • Scalene Triangle: All the sides and angles of this triangle are of different length.
  • Isosceles Triangle: Two of the three sides of a triangle are equal. And also angles opposite equal angles are equal. 
  • Equilateral Triangle: All the sides and angles of this triangle are equal.
Some Other Properties of Triangles
1. Sum of any two sides of a triangle is always greater than the third side.

2. Difference of any two sides of a triangle is always less than the third side.

3. Greater angle has greater side opposite to it and the smaller angle has smaller side opposite to it.

4. Let p, q and r be the three sides of a △ ABC and r is the largest side. Then
  • If r² < p² + q², the triangle is acute angle triangle
  • If r² = p² + q², the triangle is right angled triangle 
  • If r² > p² + q², the triangle is obtuse angle triangle
Important Terms Associated With Triangles

1. Altitude / Height and Orthocentre 

The perpendicular drawn from the opposite vertex of a side in a triangle is called an altitude or height.
The point of intersection of these three altitudes is called orthocentre. 

In this triangle AE, CD and BF are the Altitudes and O is the orthocentre. 

∠BOC = 180° - ∠BAC
∠COA = 180° - ∠ABC
∠AOB = 180° - ∠ACB

2. Median and Centroid

The line segment joining the mid-points of a side to the vertex opposite to the side is called a median.
In this triangle AE, CD and BF are the medians. The point of intersection of these three medians is called centroid.

A median bisects the area of a triangle i.e, ar (△ABE) = ar (△AEC) = ar (½ △ABC)

A centroid divides each median into the ratio of 2:1

3. Angle Bisector and Incentre

A line segment which originates from a vertex and bisects the same angle is called an angle bisector.
∠BAE = ∠CAE = ½ ∠BAC

The point of intersection of these angles bisectors is called incentre.

In this triangle AE, CD and BF are the angle bisectors and O is the incentre.

4. Perpendicular Bisector and Cirumcentre

A line segment which bisects a side perpendicularly is called at a perpendicular bisector of a side of triangle.
The point of intersection of these perpendicular bisectors is called circumcentre.

DO, EO, FO are the perpendicular bisectors in the given triangle and O is the circumcentre.

OA = OB = OC (circumradius)

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