DEFINITION OF A TRIANGLE

A triangle is a two-dimensional closed polygon having three sides and the sum of all three angles as 180 degrees.

TYPES OF TRIANGLES

For the classification of triangles according to their angles, we measure each of its interior angles and the triangles can be classified by angles, as follows:

• Acute Angled Triangle: An acute triangle has all interior angles acute (less than 90°)
• Right Angled Triangle: A right triangle has one right angle (equal to 90°)
• Obtuse Angled Triangle:  An obtuse triangle has one obtuse angle (greater than 90°)

Classification according to sides

For classification of the triangles according to their sides, we measure the length of each side of the triangle and the triangles can be classified by their sides, as follows:

• Equilateral Triangle: A triangle having all three sides of the same length is known as an equilateral triangle. Since all three sides are of the same length, all the three angles will also be equal. All interior angles of an equilateral triangle are equal to 60°

• Isosceles Triangle: A triangle having two sides of the same length and the third side of a different length is an isosceles triangle.

The angles opposite to the equal sides are equal.

• Scalene Triangle: A triangle having all three sides of different lengths is a scalene triangle.

Due to the three sides having different lengths, the three angles will also be of different measurements.

Classification according to angles and sides

For the classification of triangles according to both angles and sides, we measure the interior angles and length of the sides of the triangle. The triangles can be classified as follows:

Special cases of right-angled triangles

45-45-90 triangle

In such a triangle,

• Two angles measure 45°, and the third angle is a right angle.
• The sides of this triangle will be in the ratio – 1: 1: √2 respectively.
• This is also called an isosceles right-angled triangle since two angles are equal

30-60-90 triangle

In such a triangle,

• One angle = 90°
• The angles of this triangle are in the ratio – 1: 2: 3
• The sides opposite to these angles will be in the ratio – 1: √3: 2 respectively
• It is classified as a scalene right-angled triangle since all three angles are different.

Properties of triangle

• Angle sum property

The sum of all internal angles of a triangle is always equal to 180°. This property is called the angle sum property of a triangle.

The sum of all exterior angles of any triangle is equal to 360°

Example:

Note:- The sum of the length of any two sides of a triangle is greater than the length of the third side.

The side opposite to the largest angle of a triangle is the largest side.

Exterior Angle Property

Any exterior angle of the triangle is equal to the sum of its interior opposite angles. This property is called the exterior angle property of a triangle.

Perimeter of a triangle

The perimeter of any two-dimensional figure is defined as the distance around the circumference of the figure. It can be calculated by the perimeter of any closed shape by adding the length of each of the sides.

To summarise, the sum of the lengths of the sides is the perimeter of any polygon.

Perimeter = Sum of the three sides

Perimeter, P = a + b +c

Perimeter of an Isosceles, Equilateral and Scalene Triangle

Example 1: Find the perimeter of a triangle whose each side is 15 cm.

Solution: Since all three sides are equal in length, the triangle is an equilateral triangle;

a = b = c = 15 cm

Perimeter = a + b + c

= 15 + 15 + 15

= 45

Perimeter = 45 cm. 

Example 2: What is the missing side length of a triangle whose perimeter is 50 cm and two sides are 15 cm each?

Solution: Given,

Perimeter = 50 cm

The length of the two sides is the same i.e. 15 cm.

(Therefore, the triangle is an isosceles triangle.)

Using formula: 

P = 2l + b

50 = 2 * 15 + b

50 = 30 + b

or b = 20

Missing side length is 20 cm.

Semi-perimeter of a triangle

The semi perimeter of a triangle is half the sum of all its sides. The semi perimeter of a triangle can be calculated by dividing the perimeter of the triangle by two. 'Semi' means half, hence it can be calculated as mentioned before.

Additionally, by prior knowledge, we know that the perimeter of a triangle is the sum of the lengths of all its sides. Considering, a triangle with side lengths 'a', 'b', and 'c', the perimeter can be calculated as, 

Perimeter = a + b + c.

Using this formula, the formula for the semi perimeter of the triangle will become,

Semi perimeter of triangle = (a + b + c)/2

Example 1: Find the semi perimeter of an equilateral triangle that has a side length of 20 units.

Solution:

Since it is an equilateral triangle, all three sides are of equal measure. This means the value of the sides of the triangle can be written as: 'a' = 20 units, 'b' = 20 units, and 'c' = 20 units

We will use the formula for the semi perimeter of a triangle, s = (a + b + c)/2

Substituting the values of 'a', 'b', and 'c', Semi perimeter (s) = (20 + 20 + 20)/2 = 60/2 = 30 units.

Answer: Therefore, the semi perimeter of the equilateral triangle is 30 units.

Example2: Find the semi perimeter of a triangle with sides 5 units, 4 units, and 3 units.

Solution: The sides of the triangle are given as 'a' = 5 units, 'b' = 4 units, and 'c' = 3 units

We will use the formula for the semi perimeter of triangle, s = (a + b + c)/2

Substituting the values of 'a', 'b', and 'c', Semi perimeter (s) = (5+4+3)/2 = 12/2 = 6 units.
Answer: The semi perimeter of the triangle is 6 units.

Area of a triangle

The area of a triangle is the total space occupied by the three sides of a triangle in the 2-dimensional plane. 

The basic formula for the area of a triangle is equal to half the product of its base and height, 

A = 1/2 × b × h. 

The above formula is applicable to all types of triangles, may it be a scalene triangle, an isosceles triangle, or an equilateral triangle. Note that the base and the height of a triangle are perpendicular to each other.

Area of an equilateral triangle

An equilateral triangle is a triangle having all the sides equal. The perpendicular drawn from the vertex of the triangle to the base divides the base into two equal parts. The area can be calculated as below;

Area of an Equilateral Triangle = A = (√3)/4 × side2)

Area of a right-angled triangle

In a right-angled triangle, the height of the triangle is the length of the perpendicular side, thus the area can be written as;

Area of a Right Triangle = A = 1/2 × Base × Height

Area of an isosceles triangle

An isosceles triangle is a triangle having two equal sides and the angles opposite the equal sides are also equal.

Area of an Isosceles Triangle = A = (b(√4a2−b2))/4

where 'b' is the base and 'a' is the measure of one of the equal sides.

Area using heron’s formulae

When the length of the 3 sides of the triangle is known, the heron’s formula is used to calculate the area of the triangle. Thus, we need to know the perimeter of the triangle (which is the distance covered around the circumference of the triangle and is calculated by adding the length of all three sides). The Heron’s formula has two steps:

Step 1: Find the semi perimeter of the given triangle by adding all three sides and dividing it by 2.

Step 2: Use the value of the semi-perimeter of the triangle in the formula below, called 'Heron’s Formula'.

Area = √s(s−a)(s−b)(s−c)s(s−a)(s−b)(s−c)

Note:- (a + b + c) is the perimeter of the triangle. Therefore, 's' is the semi-perimeter which is: (a + b + c)/2

Example 1: Find the area of a triangle with a base of 10 inches and a height of 50 inches.

Solution:

Let us find the area using the area of triangle formula:

Area of triangle = (1/2) × b × h

A = 1/2 × 10 × 50

A = 1/2 × 500

Therefore, the area of the triangle (A) = 250 in2

Example 2: Find the area of a triangle with a base of 8 cm and a height of 12 cm.

Solution:

Area of triangle = (1/2) × b × h

A = 1/2 × 8 × 12

A = 1/2 × 96

A = 48 cm2

Example 3: Find the area of an equilateral triangle with a side of 4 cm.

Solution:

We can calculate the area of an equilateral triangle using the area of triangle formula, Area of an equilateral triangle = (√3)/4 × side2

where 'a' is the length of one equal side. On substituting the values, we get, Area of an equilateral triangle = (√3)/4 × 42

Answer= 6.29 cm2

Similar Triangles

Similarity Criterion

• AAA Similarity (Angle-Angle-Angle) 
• SAS Similarity (Side-Angle-Side)        
• SSS Similarity (Side-Side-Side)           
• RHS Similarity (Right Angle- Hypotenuse-Side)

1. AAA Similarity (Angle-Angle-Angle)

The above criteria state that two triangles are similar if the pairs of corresponding angles are equal.

∠A = ∠P, ∠B = ∠Q ∠C = ∠R

2. SAS Similarity (Side-Angle-Side)        

The second theorem requires an exact order: aside, then the included angle, then the next side. The above theorem states that if two sides of one triangle are proportional to two corresponding sides of another triangle, and their corresponding included angles are congruent, the two triangles are similar.

∠A = ∠P AB/PQ = AC/PR

3. SSS Similarity (Side-Side-Side)

This theorem states that two triangles are said to be similar if the sides of a triangle are proportional to the sides of the other triangle. 

AB/PQ = BC/QR = AC/PR

4. RHS Similarity (Right Angle- Hypotenuse-Side)

This theorem states that two right-angled triangles are said to be similar if the hypotenuse and one side of one triangle are proportional to the corresponding hypotenuse and one side of the other triangle

 ∠C = ∠R =90’ AB/PQ = BC/QR 

• PYTHAGORAS THEOREM

According to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of the squares of the remaining two sides of the triangle. 

For triangle ABC  we have;

BC2 = AB2 + AC2

Example 1: The hypotenuse of a right-angled triangle is 10 units and one of the sides of the triangle is 8 units. Find the measure of the third side using the Pythagoras theorem formula.

Solution:

Given : Hypotenuse = 10 units
Let us consider the given side of a triangle as the perpendicular height = 8 units
On substituting the given dimensions to the Pythagoras theorem formula
Hypotenuse2 = Base2 + Height2
102 = B2 + 82
B = √36 = 6 units
Therefore, the measure of the third side of a triangle is 6 units