# Triangles

Sub Topics

Triangle is a two dimensional figure obtained by joining 3 line segments. It is a polygon with three sides.

A triangle has 3 sides, 3 angles and 3 vertices. The symbol used to represent triangle is $\Delta$ (Delta). The below figure represents a triangle.

The triangle is denoted as $\Delta$ABC

In the above triangle ABC, A, B and C are the three vertices of the triangle.

AB, BC and AC are the sides of the triangle.

$\angle$ BAC, $\angle$ ABC and $\angle$ ACB are the three angles of the triangle.

## Properties of Triangles

Perimeter: The sum of all the sides of the triangle is called Perimeter.

Sum of the Angles:
The sum of all the angles of the triangle is always equal to 180 degrees.

Consider the following figure,

In the above figure,

Perimeter P = Side AB + Side BC + Side AC

Sum of the angles, Angle BAC + Angle ABC + Angle ACB = 180 degrees

Area of a triangle:

Area of a triangle is given by the product of $\frac{1}{2}$, base and height.

Area = $\frac{1}{2}$ * Base * Height

Where, any side can be considered as Base and Height is the perpendicular distance of the base to the opposite vertex.

The below diagram explains the area of the triangle.

In the above $\Delta$ XYZ, YZ is considered as the base. Therefore the perpendicular distance of the base YZ from the opposite vertex X is XM, i.e., XM is the Height of the triangle XYZ

Therefore Area of the $\Delta$ XYZ = $\frac{1}{2}$ * Base * Height

Area = $\frac{1}{2}$ * YZ * XM

Area = $\frac{1}{2}$ * B * H

Triangle is a two dimensional figure, it does not have volume.

## Types of Triangles

There are different types of triangles discussed below.

Equilateral Triangles: Equilateral Triangle is a triangle in which the lengths of all the sides of the triangle are equal. If the sides are equal, then the angles opposite to them will also be equal. Therefore, in an equilateral triangle, all the angles are also equal and equal to $\frac{180}{3}$ = 60 degrees.

The figure below shows the diagram of an equilateral triangle.

In the above equilateral triangle, Side AB = Side BC = Side AC

Angle ABC = Angle ACB = Angle BAC = 60 degrees (because sum of all the angles of a triangle is equal to 180 degrees).

Isosceles Triangle: In an Isosceles triangle, only two of the sides of the triangle are equal. The length of the third side is not equal to the other two sides. Therefore, in any isosceles triangle, two are always equal.

Consider the figure below:

In the above Isosceles triangle XYZ, Side XY = Side XY, and Angle XYZ = Angle XZY.

Scalene Triangle: In a scalene Triangle, all the sides of a triangle are unequal. Therefore no angles in a scalene triangle are equal. The following figure represents a scalene triangle.

In the above scalene triangle MNO, MN $\neq$ MO $\neq$ ON and

Angle MNO $\neq$ Angle MON $\neq$ Angle OMN.

## Triangle Inequality

Triangle Inequality states that the sum of lengths of any two sides of triangle is always greater than the length of the third side. Consider the below figure,

In the above $\Delta$ ABC,

a is the length of side BC,

b is the length of side AC and

c is the length of side AB.

So According to Triangle Inequality, a < b + c,

b < a + c and

c < a + b

Perimeter is the sum of the lengths of all the three sides of the triangle.

P = a + b + c

Therefore once we are given Perimeter and lengths of any two sides of a triangle, we can find the length of the third side of the triangle using the above relations and once the lengths of the sides are given, we can determine whether the sides can form a triangle or not based on the above relations.

If the length of one of the sides is equal to the sum of the lengths of the other two sides, then the figure becomes a straight line.