 # Properties of Triangle

We will discuss the properties of triangle here along with its definitions, types and its significance in Maths. A triangle definition states it is a polygon that consists of three sides, three edges, three vertices and the sum of internal angles of a triangle equal to 1800. Depending upon the sides and angles of a triangle, we have the different types of triangles, which we will discuss here. Triangle is an important concept which is taught in most of the classes like Class 7, Class 8, Class 9, Class 10 and in Class 11.

At the beginning, we start from understanding the shape of triangles, its types and properties, theorems based on it such as Pythagoras theorem, etc. In higher classes, we deal with trigonometry, where the right-angled triangle is the base of the concept. Let us learn here some of the fundamentals of the triangle by knowing its properties.

## Types of Triangle

 Based on the Sides Based on the Angles Scalene Triangle Acute angled Triangle Isosceles Triangle Right angle Triangle Equilateral Triangle Obtuse-angled Triangle

So before, discussing the properties of triangles, let us discuss these above-given types of triangles.

Scalene Triangle: All the sides and angles are unequal.

Isosceles Triangle: It has two equal sides and angles opposite to these equal sides are also equal.

Equilateral Triangle: All the sides are equal and all the three angles are of 600.

Acute Angled Triangle: A triangle having all its angles less than 900.

Right Angled Triangle: A triangle having one of the three angles is 900.

Obtuse Angled Triangle: A triangle having one of the three angles as more than 900.

Now that you have got the details of all the triangles, let us further discuss triangle properties.

## Triangle Properties

The properties of the triangle are:

• The sum of all the angles of a triangle(of all types) is equal to 1800.
• The sum of the length of the two sides of a triangle is greater than the length of the third side.
• In the same way, the difference between the two sides of a triangle is less than the length of the third side.
• The side opposite the greater angle is the longest side of all the three sides of a triangle.
• The exterior angle of a triangle is always equal to the sum of the interior opposite angles. This property of a triangle is called an exterior angle property
• Two triangles are said to be similar if their corresponding angles of both triangles are congruent and lengths of their sides are proportional.
• Area of a triangle = ½ × Base × Height
• The perimeter of a triangle = sum of all its three sides

### Triangle Formula

• Area of triangle is the region occupied by a triangle in a two-dimensional plane. The dimension of area is square unit. The formula for area is given by;

Area = 1/2 x Base x Height

• The perimeter of a triangle is the length of the outer boundary of a triangle. To find the perimeter of a triangle we need to add the length of the sides of the triangle.

P = a + b + c

• Semi-perimeter of a triangle is half of the perimeter of the triangle. It is represented by s.

s = (a + b + c)/2

where a , b and c are the sides of the triangle.

• By Heron’s formula, the area of the triangle is given by:

A = √[s(s-a)(s-b)(s-c)]

where ‘s’ is the semi-perimeter of the triangle.

• By Pythagorean theorem, the hypotenuse of a right-angled triangle can be calculated by the formula:

Hypotenuse2 = Base2 + Perpendicular2

## Properties of Triangle Examples

Example 1: If an equilateral triangle has lengths of sides as 5 cm and perpendicular is drawn from the vertex to the base of the triangle. Then find its area and perimeter.

Solution: Given, side of the equilateral triangle, say ABC = 5 cm

If we draw a perpendicular from the vertex of an equilateral triangle, A to the base at point O, it divides the base into two equal sides. Such that, BO = OC = 2.5 cm

Now, the area of triangle = ½ × Base × Height

To find the height of the triangle, AOB, we have to use Pythagoras theorem.

That is, Hypotenuse2 = Base2 + Perpendicular2

Or Perpendicular = $\sqrt{Hypotenuse^2-Base^2}$

Therefore, OA = $\sqrt{AB^2-OB^2}$

Or OA = $\sqrt{5^2-2.5^2}$

OA = $\sqrt{25-6.25} = \sqrt{18.75}$

Area of triangle = ½ × OB × OA

= ½ × 2.5 × $\sqrt{18.75}$ = ½ × 2.5 × 4.33

Area of triangle ABC = 5.4125 cm2

Perimeter of triangle ABC = sum of all its three sides

= 5+5+5 cm

= 15cm

Example 2: If the sides of a scalene triangle are given by 3cm, 4cm and 5 cm, where base is 4cm and the altitude of the triangle is 3.2 cm. Then find the area and perimeter of the triangle.

Solution: Let the sides of the triangle are:

a = 3cm, b = 4cm and c = 5cm

Altitude is the height of the triangle = 3.2cm

By the formula of area of the triangle, we know;

Area = 1/2 x base x height

A = 1/2 x 4 x 3.2

A = 6.4 sq.cm.

Now, the perimeter of the triangle is given by;

P = a + b + c

P =  3+4+5

P = 12cm.

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