# Equilateral Triangle

## Definition

A triangle whose lengths of three sides are equal, is called equilateral triangle.

The lengths of three sides can be equal in some special cases and it is called as equilateral but all three equal length sides are part of the triangle. So, the triangle is known as equilateral triangle geometrically.

An equilateral triangle is symbolically represented by displaying a perpendicular minor line to each side at its middle point.

### Properties

The $\Delta EFG$ is a basic example to study the properties of the equilateral triangle.

#### Sides

$\overline{EF}$, $\overline{FG}$, and $\overline{GE}$ are three sides of the triangle $EFG$ and lengths of them are equal.

$\therefore \,\, EF = FG = GE$

Due to having equal length sides, the $\Delta EFG$ is called as an equilateral triangle.

#### Angles

An equilateral triangle has three interior angles but equiangular triangle due to having equal length sides.

The summation of all three angles is $180^\circ$ in a triangle and it is equally divided into three equal angles due to three equal length sides. Therefore, each interior angle is $60^\circ$ in an equilateral triangle.

$\angle GEF = \angle EFG = \angle FGE = 60^\circ$

##### Construction

An equilateral triangle can be constructed geometrically by using geometrical tools to study its properties.

1. Take ruler and draw a straight line of any length in horizontal direction. As an example, $10$ centimeters horizontal straight line is drawn. The left and right side endpoints are called as point $M$ and point $N$.
2. Take compass and also take help of ruler to set the distance of $10$ centimeters between needle point and pencil lead point.
3. Now draw an arc above the line from point $M$ and also draw another arc over the line from point $N$. The two arcs drawn from points $M$ and $N$ cut each other at a point and the point is called as point $O$.
4. Take ruler and join the points $M$ and $O$ by drawing a straight line, and then join the points $N$ and $O$ in same way.

Thus a triangle $MNO$ is constructed geometrically. The points $M$, $N$ and $O$ are vertices of the triangle. The line segments $\overline{MN}$, $\overline{MO}$ and $\overline{ON}$ are sides of the triangle.

1
###### Lengths of sides

Now measure the lengths of all three sides by using ruler.

1. Length of side $\overline{MN}$ is $MN = 10$ centimetres
2. Length of side $\overline{MO}$ is $MO = 10$ centimetres
3. Length of side $\overline{ON}$ is $ON = 10$ centimetres

The lengths of all three sides are equal and length of the each side is $10$ centimetres.

$MN = MO = ON = 10$ Centimeters

Therefore, $\Delta MNO$ is called as an equilateral triangle.

2
###### Interior angles

After that take protractor and measure each interior angle of the triangle.

1. The interior angle $\angle NMO = 60^\circ$
2. The interior angle $\angle MON = 60^\circ$
3. The interior angle $\angle ONM = 60^\circ$

All three interior angles are equal and each interior angle is $60^\circ$.

$\angle M = \angle N = \angle O = 60^\circ$

Having equal lengths of all three sides is the main reason for the equiangular in equilateral triangle.

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