# Triangles

A triangle is a polygon of three sides. Sum of the angles of a triangle is 180 degrees. Triangles are classified in two general ways: by their sides and by their angles.

### Types of Triangle

Based on sides, triangles are classified into three categories.

**1. Scalene triangle**

A triangle with three sides of different lengths is called a scalene triangle.

**2. Isosceles triangle**

An isosceles triangle has two equal sides. The third side is called the base. The angles that are opposite to the equal sides are also equal.

**3. Equilateral triangle**

An equilateral triangle has three equal sides. In this type of triangle, the angles are also equal, so it can also be called an equiangular triangle. Each angle of an equilateral triangle must measure 60°, since the sum of the interior angles of any triangle must equal to 180°.

Triangles are also divided into three classes on the basis of measure of the interior angles.

**Obtuse angled triangle**

When the measure of the largest angle of the triangle is greater than 90° then it is an obtuse angled triangle. In the figure ∆ABC is an obtuse triangle where C is an obtuse angle.

**Acute angled triangle**

All angles are less than 90°. For example, ∆PQR is a acute triangle because largest angle is less than 90°.

**Right angled triangle**

A triangle whose one angle is 90° is called a right (angled) triangle. In figure, b is the hypotenuse, and a & c the legs, called base and height respectively. In right triangle ABC,

a^{2} + b^{2} = c^{2}

### Properties of Triangle

- The sum of the three angles is 180°.
- The sum of any two sides of a triangle is greater than third side.
**Pythagoras Theorem:**In a right angled triangle (Hypotenuse)^{2}= (Base)^{2}+ (Height)^{2}- The line joining the mid point of a side of a triangle to the opposite vertex is called the Median.
- The point where the three medians of a triangle meet, is called Centroid. The centroid divides each of the medians in the ratio 2:1.
- In an isosceles triangle, the altitude from the vertex bisects the base.
- The median of a triangle divides it into two triangles of the same area.
- The area of the triangle formed by joining the mid points of the sides of a given triangle is one-fourth of the area of the given triangle.

### Centroid

The point of intersection of the **medians** of a triangle. Median is the line joining the vertex to the mid-point of the opposite side.

The centroid divides each median from the vertex in the ratio 2 : 1.

To find the length of the median, use the theorem of Apollonius.

AB^{2} + AC^{2} = 2(AD^{2} + BD^{2})

The medians will bisect the area of the triangle.

If x, y, z are the lengths of the medians through A, B, C of a triangle ABC, then “Four times the sum of the squares of medians is equal to three times the sum of the square of the sides of the triangle”.

4(x^{2} + y^{2} + z^{2}) = 3(a^{2} + b^{2} + c^{2})

### Orthocentre

This is the point of intersection of the **altitudes**. Altitude is a perpendicular drawn from a vertex of a triangle to the opposite side.

In a right angled triangle, the orthocenter is the vertex, where the right angle is.

### Circumcentre

It is the point of intersection of **perpendicular bisectors** of the sides of the triangle.

The Circumcentre of a triangle is the centre of the circle passing through the vertices of a triangle.

The Circumcentre is equidistant from the vertices.

If a, b, c, are the sides of the triangle, ∆ is the area, then

abc = 4R∆

where R is the radius of the circum-circle.

### Incentre

This is the point of intersection of the **internal bisectors** of the angles of a triangle.

∆ = rs if r is the radius of incircle, where s = semi-perimeter = (a + b + c)/2 and ∆ is the area of the triangle.

BF = BD = s - b where 2s = a + b + c, CE = CD = s - c. AF = AD = s - a

The angle between the internal bisector and the external bisector is 90°.

### Equilateral Triangle

In an equilateral triangle all the sides are equal and all the angles are equal.

**1. Altitude** = √3/2 × side = √3a/2

**2. Area** = √3/4 × (side)^{2} = √3a^{2}/4

**3. Inradius** = 1/3 × Altitude

**4. Circumradius** = 2/3 × Altitude