1. Triangles and their types
1.1 What is a triangle?
A triangle is a 2D shape with three sides. It has three angles and three vertices.
1.2 Types of triangles
Based on length, there are three types of triangles:
a) Equilateral Triangle
An equilateral triangle has all three sides of equal length. Also, all the three angles
are equal and measure 60° each.
b) Isosceles Triangle
An isosceles triangle has two sides of equal length. Angles opposite to two equal sides are also equal
c) Scalene Triangle
A scalene triangle has all three sides of different lengths. All three angles are also of different measures.
What are Right-angled triangles?
A triangle in which one of the angles measures 90° is called a right-angled triangle.
A right-angled triangle has one right angle and two acute angles.
1.3 Inequality Rule
The sum of lengths of any two sides of a triangle must be greater than the third side.
For example:
According to the rule, a + b > c
b + c > a
a + c > b
Example: Is it possible to form a triangle with the following measures?
12 cm 15 cm 8 cm
Solution:
Let a=12 cm
b=15 cm
c=8 cm
According to the Inequality rule,
a + b > c
a + b = 12 + 15 = 27
27 > 8
b +c>a
b + c = 15 + 8 = 23
23 > 12
c + a > b
c + a = 10 + 9 = 19
19 > 12
As the measures given follow the inequality rule, they can form a triangle.
2. Angles in triangles
A triangle has three angles. Doesn’t matter, which type of triangle it is, the sum of all the three angles in a triangle will always be equal to 180 degrees.
The above triangle has three angles a, b, and c. So a + b + c = 180°.
2.1 Example Question on working out missing angles in triangles.
For the triangle shown below, find the value of x.
Solution:
Given triangle has one right angle, so it's a right-angled triangle.
So one angle is 90°.
Other acute angle is 50°.
Sum of all the angles in a triangle is 180°.
x + 90° + 50° = 180°
x + 140° = 180°
x = 180° - 140°
x = 40°
2.2 Example Question on missing angles in an isosceles triangle
Work out the value of the angle marked x for the isosceles triangle shown below.
Solution:
For an isosceles triangle, angles opposite to equal sides are equal.
The Sum of all the angles in a triangle is 180°.
x + x + 65° = 180°
2x + 65° = 180°
2x = 180° - 65° = 115°
x = 115°÷ 2
x = 57.5°
3. Perimeter and area of triangles
Perimeter of a triangle: Sum of all the sides of a triangle.
Let us see the same triangle we used above.
The perimeter of the above triangle is a + b + c
Formula for Area of Triangle:
Area of a triangle can be calculated by multiplying its height by the base and dividing by 2.
Its formula is Area of triangle = 1⁄2 × base × height
Line joining one of their sides (called the base) with the opposite point at 90 degrees is known as height.
3.2 Solved Example
Calculate the perimeter and the area of the triangle shown below.
Solution:
Perimeter of a triangle = Sum of lengths of all three sides
=7 + 9 + 4
= 20 cm
Area of a triangle = 1⁄2 × b× h = 1⁄2 × 9 × 3
= 13.5 cm²
