Triangle - Types Of Triangles with Properties Of A Triangle

Introduction to triangle

A triangle is a closed curve which is formed by 3-line segments. The line segments by which the triangle is formed are called sides of the triangle. The points of intersections of the sides of the triangle are called the vertices of the triangle. The angles formed at the vertices are called the angles of the triangle.

Properties of a Triangle

The properties of the triangle are:

• The sum of all the angles of a triangle (of all types) is equal to 180°.
• 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 the lengths of their sides are proportional.
• Area of a triangle = ½ × Base × Height
• The perimeter of a triangle = sum of all its three sides

What Are the different Types of Triangles?

a. Right Triangles

In a right triangle, one of the angles is equal to 90° or right angle.

b. Equilateral triangle

An equilateral has all three sides equal to each other. Due to this all the internal angles are of equal degrees, that is, each of the angles is 60°

c. Isosceles triangle

In an isosceles triangle, two sides have equal length. The two angles opposite to the two equal sides are also equal to each other.

d. Scalene Triangle

A scalene triangle is a type of triangle, in which all the three sides have different side measures. Due to this, the three angles are also different from each other.

e. Acute Triangle

An acute triangle has all of its angles less than 90°.

f. Obtuse Triangle

An obtuse triangle has any of its one angles more than 90°.

How do classify a triangle?

Classifying Triangles by Sides

To classify a triangle by its sides means that we look at the side lengths of the triangle and make a determination as to whether it is an: Equilateral, Isosceles and Scalene. To be an equilateral triangle all three-side length must be exactly the same. An Isosceles triangle will have at least 2 side lengths that are the same. If all three sides of the triangle are different then the triangle is scalene. Classifying a triangle is as simple as comparing the sides. If all three sides have the same length, then it is an equilateral triangle, if only two sides have the same length, then it is an isosceles triangle and if there are no sides that have the same length then it is a scalene triangle.

1.If all the sides are equal (the same length) then the triangle is Equilateral.

2. If 2 sides of the triangle are the same length then the triangles are an Isosceles triangle.

3. If all three sides of the triangle are a different length, then the triangle is a Scalene triangle.

Classifying Triangles by Angles

There are two ways to classify triangles. To classify a triangle by angles means to categorize the triangle according to the types of angles that make up the triangle. To classify triangles by angles you must determine if each angle in the triangle is acute, right or obtuse. Once you have determined the types of angles you can classify the triangle as an Acute Triangle, Right Triangle, or Obtuse Triangle.

1. If all the angles of the triangle are all less than 90° then the triangle is classified as an Acute Triangle. If all the angles in the triangle are 60° or if they are all marked as congruent then the triangle is an acute triangle but it is also a special triangle called an equiangular triangle.

       

   2. If one of the angles is 90° then the triangle is a Right Triangle.

   3. If one of the angles is greater than 90° then the triangle is classified as an Obtuse Triangle.

What is Pythagorean Theorem & Formula  of a triangle

Pythagoras Theorem is an important topic in Maths, which explains the relation between the sides of a right-angled triangle. It is also sometimes called the Pythagorean Theorem. Pythagoras theorem is basically used to find the length of an unknown side and angle of a triangle. By this theorem, we can derive base, perpendicular and hypotenuse formula. Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides “. The sides of this triangle have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple.

The theorem is named after a Greek Mathematician called Pythagoras.

Pythagoras Theorem Formula

Consider the triangle given above:

Where “a” is the perpendicular,

“b” is the base,

“c” is the hypotenuse.

According to the definition, the Pythagoras Theorem formula is given as:

Hypotenuse² = Perpendicular² + Base² 

c² = a² + b²  

The side opposite to the right angle (90°) is the longest side (known as Hypotenuse) because the side opposite to the greatest angle is the longest.

Consider three squares of sides a, b, c mounted on the three sides of a triangle having the same sides as shown.

By Pythagoras Theorem –

Area of square “a” + Area of square “b” = Area of square “c”