Triangles are classified in terms of their sides and the angles.

**Triangles classified according to their sides are, **

1. Scalene Triangle.

2. Isosceles Triangle

3. Equilateral Triangle

**Triangles classified according to their angles are,**

1. Acute angled triangle.

2. Obtuse angled triangle.

3. Right angled triangle.

### A. **Triangles classified according to their sides**

**Scalene Triangle:** A triangle in which no sides are equal is called a scalene triangle.

The following figure shows the shape of a scalene triangle.

For a given scalene triangle, we can find the area using the formula

Area =

$\frac{1}{2}$ . base $\times$ height

The above figure shows the scalene triangle whose base and the height are known.

If the length of the sides of the triangle are known we can find the area using

**Heron's formula.**The formula can be given by

** $\sqrt{s(s-a)(s-b)(s-c)}$ **where s is the semi perimeter given by

**s = $\frac{a+b+c}{2}$**For a given scalene triangle if we are given a pair of adjacent sides and the angled included by the two sides are given,

we can find the area using the formula given below.

In the above figure the given adjacent sides are AC and AB and included angle is $\angle CAB$

** Area = $\frac{1}{2}$ AB x AC sin $\angle CAB$**If we are given a pair of angles of a triangle and one of the side opposite to the given angle we can find the other sides using

the

**Law of Sines** given by

**$\frac{a}{sin\angle A}$ = $\frac{b}{sin\angle B}$ = $\frac{c}{sin\angle C}$**

We can also solve for remaining sides and the angles of a scalne triangle using Law of cosines given by,

** $a^{2}$ = $b^{2}$ + $c^{2}$ - 2 bc cos $\angle A$** ** $b^{2}$ = $c^{2}$ + $a^{2}$ - 2 ac cos $\angle B$**

$c^{2}$ = $a^{2}$ + $b^{2}$ - 2 ab cos $\angle C$**Isosceles Triangle:** A triangle whose two of its sides are equal is called an isosceles triangle.

In the following figure, the sides AB = AC.

**Isosceles triangle Theorem: 1** If two sides of a triangle are equal then the angles opposite to them are equal.

In the above triangle, AB = AC => $\angle B$ = $\angle C$

**Converse of Isosceles Triangle Theorem:** If two angles of a triangle are equal, then the sides opposite to

equal angles of the triangle are equal.

If $\angle B$ = $\angle C$ => AB = AC

**Isosceles Triangle Theorem 2: **The perpendicular drawn from the vertex of an isosceles triangle bisects the third side.

**Isosceles Triangle Theorem 3: **The median drawn to the third side of an isosceles triangle is perpendicular to the base (third side).

**Isosceles
Triangle Theorem 4:** The angular bisector of the vertex of an isosceles
triangle divides the triangle into two congruent

right triangle.

**Area of the isosceles triangle: **We can find the area of an isosceles traingle as using the formula,

** Area = $\frac{1}{2}$ x Base x Height ** (or)

** Area = $\frac{1}{2}$ BC x AB sin $\angle ABC$****Equilateral Triangle: **A triangle whose sides are of equal measures is called an equilateral triangle.

In an equilateral triangle,

**each angle measure 60**^{o}The following diagram shows an equilateral triangle whose sides are equal to 2a units.

The

**area of an equilateral triangle = $\frac{\sqrt{3}}{4}$ a**^{2}**,** where 2a is the length of the side.

### B. **Triangles classified according to their angles **

**Acute Triangles (also called Acute angled triangle): **A triangle whose each of the angle measure less than 90

^{o} (acute angle) is called an Acute Triangle.

In the above $\Delta ABC$, each of the angles A, B and C are acute.

**Right Triangles (also called a right angled triangle): **A triangle in which one of the angle measure exactly 90

^{o} is called a Right Triangle.

**Obtuse Triangle (also called obtuse angled triangle): **A triangle in which one of the angle measure more than 90

^{o} (obtuse angle) is called an obtuse angled triangle.

IT is not possible for a triangle to have more than one obtuse angle.In the above triangle $\angle PQR$ is obtuse.

An obtuse angled triangle can be scalene or isosceles, but can never be an equilateral triangle

Area of an obtuse angled triangle =

$\frac{1}{2}$ x Base x Height

Height of the triangle will be outside the triangle if one of the side containing the obtuse angle is a base.