Be it the great pyramids of Egypt or a tempting piece of Pizza, the shape of a triangle has never really failed to fascinate us with its utility and its presence in our day to day life.

What is a triangle?

Ideally, a triangle is a 3-sided flat shape with straight sides. You connect three lines and voila, you have created one of the strongest shapes that exist.
It is also a polygon (In geometry, a polygon can be defined as a flat, closed shape with straight sides). We can find the shape of a triangle in a flag or even in a roadside signboard.

The word Triangle can be simply broken down into two parts, ‘tri’ and ‘angle’.
The English prefix tri that has been derived from both Greek and Latin, means “three”.
As one can figure out, the triangle consists of three angles (the space usually measured in degrees, between two intersecting lines). It has three sides and three vertices (typically means a corner or a point where lines meet).

Properties of a triangle

A triangle with vertices A, B and C is denoted as △ABC.
• The sum of the three interior angles of a triangle is 180°. ∠A + ∠B + ∠C = 180°.
• The sum of an interior angle of a triangle and its corresponding adjacent angle is always 180°. Hence ∠BAC + ∠HAE = 180°, ∠ACB + ∠FCH = 180° and ∠ABC + ∠IBG = 180°.
• The sum of the length of two sides of a triangle is always more than the length of the third side. Hence AB + BC > AC, AB + AC > BC and AC + BC > AB.
• The two exterior angles of a triangle having the same vertex are congruent (Congruent angles have the exact same measure). In simple terms this applies since they are vertically opposite angles. Hence ∠DAG = ∠EAH, ∠DCI = ∠GCH and ∠FBE = ∠IBG
• Exterior angle theorem: The measure of an exterior angle of a triangle is always equal to the sum of the measures of the two interior angles (called remote interior angles) that are not adjacent to it. This is called as exterior angle theorem. Hence ∠FBA = ∠BAC + ∠BCA, ∠ICA = ∠ABC + ∠BAC, ∠CAE = ∠ABC + ∠ACB

Triangles example:

Q.1) If in the figure below, ∠BAC = 60° and ∠BCA = 60°, what is the measure of ∠FBA?

Solution: We know that ∠BAC = 60° and ∠BCA = 60°
We know that ∠FBA = ∠BAC + ∠BCA (Exterior angle theorem)
∴ ∠FBA = 60° + 60°
∴ ∠FBA = 120°
• The exterior angles of a triangle always add up to 360 degrees. Hence, in the figure given below, ∠A + ∠B + ∠C = 360° and ∠X + ∠Y + ∠Z = 360°.

• The shortest side is always opposite to the smallest interior angle. Similarly, the longest side is always opposite to the largest interior angle.

Types of Triangles by length of sides

• Equilateral triangle:

All edges are equal, all angles are also equal to 60°.

• Isosceles triangle:

Two edges are equal, that is why two angles are also equal or sides opposite to equal sides are equal. Therefore, an equilateral triangle having all sides equal is also an isosceles triangle.

• Scalene triangle:

All edges are different, hence all angles are correspondingly different.

Types of triangles by their internal angles

• Right Angled triangle:

A triangle with one angle equal to 90° is called right-angled triangle.

• Obtuse Triangle:

The obtuse angled triangle is the one with one obtuse angled (more than 90°) side. Isosceles triangles and scalene triangles come under this category of triangles.

• Acute Triangle:

Triangles, where all sides are acute-angled (lesser than 90°) to each other, are called acute triangles. The best example of this kind of triangle is the equilateral triangle.

Triangles example:

Q.2) Find the value of x in the triangle shown below.

Solution: we know that ∠Q= 114 and ∠P = 33°
We know that ∠P + ∠Q + ∠R = 180°
∴ x + 114° + 33° = 180°
∴ x + 147° = 180°
∴ x= 33°
∴ ∠R = 33°

Perimeter of a triangle

Definition: The sum of the measures of all three sides of a triangle is referred to as the perimeter of the triangle.

How to find the perimeter of a triangle:

The perimeter is the length of the outline of a shape. In a triangle we find out the perimeter by adding together the lengths of all the three sides. Or as a formula: Perimeter = a + b + c where a, b and c are the lengths of each side of the triangle.

Triangles example:

Q.3) The perimeter of an equilateral triangle is equal to 210 cm. What is the length of one side of this triangle?

Solution: Perimeter of an equilateral triangle = 210 cm
We know that all the sides of an equilateral triangle are equal.
Let the length of the side of the equilateral triangle be x.
∴ x + x + x = 210
∴ 3x = 210
∴ x = 210/3
∴ x = 70
∴The length of one side of this triangle is 70 cm.

Area of a triangle

The region enclosed within the three sides of a triangle is referred to as the area of the triangle.

• The area of a triangle can be found out based on the information that has been provided to us and does not always have to adhere to the standard formula that employs base and height.
• The standard or general formula for the area of the triangle is as follows: The area of a triangle is equal to half of the product of its base and height or A (△) = ½ × base × height.
Ideally, base can be any side and does not necessarily have to be just the one drawn at the bottom. To calculate the area, one must use the altitude that forms a right angle with the chosen base. An altitude of a triangle is a line segment that passes through a vertex and ends at a point on the base, thereby forming an angle of 90°. Choose any side that is convenient to be the base.
• Area of a Right Triangle = A = ½ × Base × Height (Perpendicular distance)
• Area of an Equilateral Triangle = A = (√3)/4 × side²
• Area of an Isosceles Triangle = A = ½ (base × height)

Finding the Area of Triangle with the help of Heron’s Formula

Heron's formula is named after Heron of Alexandria, a Greek engineer and Mathematician who lived in the era of 10 - 70 AD. The area of a triangle can be obtained by using Heron’s formula if all the three sides of the triangle are given. We find the semi perimeter first and then using the same in the main formula, we find the area of the triangle.

Where, s is semi-perimeter of the triangle = (a + b + c) / 2
You can use this formula to find the area of a triangle using the 3 side lengths.
The advantage with this formula is, you do not have to rely on the formula for area that uses base and height.

Triangles example:

Q.4) Find the area of an equilateral triangle whose side is 4 cm?

Solution: We know that side of the equilateral triangle = a = 4 cm
We know that the area of an equilateral triangle = (√3)/4 × side²
Area = √3 side²/ 4
= √3 (4)²/ 4
= 4√3
∴ Area of given equilateral triangle is 4√3 cm²

Triangles example:

Q.5) Find the area of a right-angled triangle whose base measures 7 cm and height equals 4 cm.

Solution: We know that area of a triangle = (½) × b × h
= (½) × (7) × (4)
= (½) × (28)
∴Area of the triangle = 14 cm²

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