## Types of Triangles With examples | Properties of Triangle

A simple closed figure bounded by three lines segments is called** triangle.**

The* triangle* can be defined as “A two dimensional plane figure with three straight sides and three angles”.

### Terminology and Formulas of the Triangles:

#### Triangle Sides :

The three lines segments that form the triangle area called sides of triangle. AB = c, BC = a, CA = b are the sides of the ΔABC ( triangle ABC). The Sum of any two sides of a triangle is always greater than the third side.

i.e a + b > c , b + c > a , c + a > b

#### Vertex :

Two adjacent sides of a triangle intersect at a point. That point is called vertex. The points A, B, &C area called vertices or angular of ΔABC ( triangle ABC) .

#### Angle :

Two lines segments with the same end point will determine one angle. Triangle has 3 angles namely

∠ABC, ∠BCA, ∠CAB and they also denoted as ∠B, ∠C, ∠A respectively. Always the sum of the angles of a triangle is

i.e ∟A + ∟B + ∟C =

#### Perimeter of a Triangle :

The sum of the measures of the three sides of the a triangle is called the** perimeter of the triangle.** The sum of the sides AB = c , BC = a & CA= b is equal to perimeter of ΔABC ( triangle ABC).

i.e perimeter of ΔABC = AB + BC + CA = a +b + c

#### Triangular Area or Region :

The interior of a triangle together with its boundary is called the triangular region or triangular area.

Area of the triangle = (1/2) x Base x Height

Area of the triangle ABC = (1/2) x a x h

#### Median of the triangle :

The line joining the midpoint of a side of a triangle to the positive vertex is called the median of the triangle. Here AR, BQ, CP are the medians of the triangle ΔABC. The median of a triangle divides the triangle into two triangles with equal areas.

i.e Area of the ΔARB = Area of the ΔARC

Area of the ΔBQA= Area of the ΔBQC

Area of the ΔCPB = Area of the ΔCPA

#### Triangle centroid:

The point where the three medians of a triangle meet is called centroid of that triangle. The point **” s”** is the centroid of the triangle ΔABC. The triangle centroid divides each the median of the triangle into the ratio segments.

i.e As : sR = 2 : 1

Bs : sQ = 2 : 1

Cs : sP = 2 : 1

Area of a triangle formed by joining the midpoints of the sides of a given triangle is one-fourth of the area of the given triangle.

Area of the triangle ABC = 1/4 area of the triangle PQR (** area of ΔABC = 0.25 x area of ΔPQR** )

### Classification of the triangles:

Triangles can be classified into three kinds according to measures of their sides and three kinds according to the measures of angles.

According to the sides the triangles classified as *Equilateral, isosceles and Scalene*

##### Equilateral triangles :

A triangle having all the three sides of the same length is called an equilateral triangle. All three angles of equilateral triangles are equal.

i.e PQ = QR = RP = a

∠P, ∠Q, ∠R = 60°

**Formulas for equilateral triangle :**

Perimeter of the equilateral triangle =** 3 a**

Area of the equilateral triangle = **(√3/4) x a ^{2}**

Radius of incircle of an equilateral triangle = **a / (2 √3)**

Radius of circumference circle of an equilateral triangle = **a / √3**

##### Isosceles Triangles :

A triangle having two sides of equal length is called an isosceles triangle. The angle opposite to the equal sides of an isosceles triangle area equal.

Here PQ = PR = a and ∠Q = ∠R

##### Scalene Triangles :

If the three sides of a triangle are of different length then the triangle is called *scalene triangle.*

If PQ ≠ QR ≠ R or ∠P ≠ ∠Q ≠ ∠R then ΔPQR is called scalene triangle.

According to their angles divided three kinds of triangles. They are acute *angled triangle, Right angled triangle and obtuse angled triangle*

##### Acute Triangle :

All angles of this triangles area having less than 90°. So all equilateral triangles area acute triangles.

If ∠P < 90° , ∠Q < 90° & ∠R < 90° than ΔPQR is called Acute triangle.

##### Right Triangle :

If triangle having one 90° angle than it is called **right angled triangle**.

If triangle having a right angle (i.e 90°) and also two equal angles ( i.e 45° & 45° ) then it is called **Right Isosceles Triangle**

Here ∠Q = 90° So ΔPQR is called right angle triangle.

If ∠P = ∠R = 45° then ΔPQR is called **Right Isosceles Triangle**.

##### Obtuse Triangle :

If triangle having one more than 90° angle than it is called Obtuse angled triangle.

Here ∠P > 90° So ΔPQR is called Obtuse angled triangle.

### Properties of the Triangles:

- The sum of the three angles of the a triangle is two right angles (i.e 180° ).
- If one sides of the triangle is produced , the exterior angle formed is equal to the sum of the interior opposite angles. Here ∠P = ∠Q = ∠PRS.
- The internal bisectors of ∠Q and ∠R of ΔPQR intersect at A. Then ∠A = 180° + (∠P / 2 ) ( Note : Bisectors means ∠AQP = ∠AQR and ∠ARP = ∠ARQ )
- If any two angles and a sides of one triangle are equal to two angles and the corresponding side of the other triangle, then the two
*triangles are congruent.* - If two medians of triangle are equal then the triangle must be an Isosceles triangles.

### Geometry Math

Two dimensional shapes formulas.

Properties of circle in math | Arc, Perimeter, Segment of circle

Quadrilateral Properties | Trapezium, parallelogram, Rhombus

Topics in Quantitative aptitude math for all types of exams

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