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Triangle and its properties

Today we are going to learn an important and fundamental topic of geometry which is the triangle. We all recognize what triangle is but few of us know the details or properties of the triangle. In this post We are going to learn triangle and its properties, its types and how to calculate area of triangle, etc. So, let’s start!

What is a triangle 

If we talk about the definition of triangle, it is nothing but a polygon having three sides and three angles.

Properties of triangle

  • The sum of all three angles of a triangle is always 180 degree. 
  • If we take the sum of the lengths of any two sides, it will be greater than the length of the remaining side.
  • If we take the difference of the lengths of any two sides, it will be less than the length of the remaining side.
  • The angle opposite the largest side of the triangle is the greatest angle and vice versa.
  • Area of a triangle = 0.5 * base * height. 
  • Area of the triangle also can be found by using heron’s formula
  • Perimeter of any triangle can be found by adding its all three sides.

In upcoming paragraphs we will see the triangle and its properties in detail with examples.

Sum of angles of a triangle

Before talking about the sum of angles of triangle we need to know what the interior and exterior angles are. The interior angles are those angles that are inside the triangle and exterior angles are those angles which are formed by one side of the triangle and the extension of the adjacent side. In the figure below, A, B, and C are the interior angles and D, E, and F are the exterior angles.

Sum of angle of a triangle

The exterior angle is equal to the sum of the interior angles of the opposite side.

For example, D = B + C.

The sum of the interior angles of a triangle is 180 degrees or π radians.

A + B + C = 180o

And Sum of the exterior angles is 360 degrees or 2π radians.

D + E + F = 360o

The side opposite to the largest angle has the longest length and the side opposite to the smallest angle has the shortest length.

Area and Perimeter of triangle

Perimeter 

The perimeter of any polygon is equal to the sum of the lengths of all sides. So, perimeter of triangle will be the sum of its three sides.

Let the length of the three sides be a, b and c, then, the triangle perimeter formula is given as:

p = a + b + c

Semi-perimeter is equal to half of the perimeter. i.e.

s = 1 / 2 (p) or s = 1 / 2 (a + b + c)

How to calculate the Area of a triangle

Let b be the base of the triangle and h be the height of the triangle. Then, the formula to calculate the area of a triangle is given as:

Δ = 1 / 2 (b × h)

The base can be the length of any side of the triangle, usually the bottom one is taken as the base. Height is obtained by drawing a perpendicular line from the base to the opposite vertex.

Heron’s formula.

The area of the triangle can also be found using only side length of the triangle. It is done using Heron’s formula.

Let a, b and c be the length of the sides of triangle and s be the semi-perimeter. Then,

Δ = √ {s(s – a)(s – b)(s – c)}

As discussed earlier semi perimeter is the half of the perimeter, i.e.

s = 1 / 2 (a + b + c) 

Types of Triangle

Triangle can be classified into various types based on sides and angles.

Classification based on sides

Based on the sides a triangle can be classified into 3 types.

  1. Equilateral triangle
  2. Isosceles triangle
  3. Scalene triangle

Equilateral Triangle

An equilateral triangle is a triangle which has all the sides of equal length. i.e.

a = b = c.

All three angles of an equilateral triangle are equal to 600 that’s why it is also known as equiangular triangle.

The sum of interior angles  = 180o

∠A +∠B + ∠C = 180o
∠A = ∠B = ∠C. Then,

3∠A = 180o
∠A = 60o

Thus, each angle is 60o. Moreover, each of the exterior angles will be equal to 120o.

Area and perimeter of equilateral triangle 
  • Area of the equilateral triangle = (√3/ 4) × a2
  • Perimeter of the equilateral triangle, p = 3a.

Where a is the length of each side

Let us practice it.

Example 1

Find the area of the equilateral triangle having the length of one side equal to 5cm.

Solution

We are only given the length of a single side. But, since it is an equilateral triangle, all the sides have lengths of 5cm. i.e.

a = b = c = 5cm.

Area of equilateral triangle = Δ
= (√3/ 4) × a2
= (√3/ 4) × 52
= (√3/ 4) × 25

= 10.825 cm2

Area of the triangle is 10.825 cm2.

Isosceles Triangle

An isosceles triangle has two sides of equal lengths. Angles corresponding to equal sides are also equal.

Scalene Triangle

A scalene triangle has all sides of different lengths.

Let us now move on to the types of triangle according to the angles.

Classification based on angles

Based on the angle a triangle can have 3 types, which are

  1. Acute triangle
  2. Obtuse triangle
  3. Right angle triangle

Acute Triangle

An acute triangle is a triangle in which all the interior angles are less than 900.

Obtuse Triangle

A triangle in which one of the angles is greater than 90o is known as an obtuse triangle.

Right-angled Triangle

A right triangle is a triangle in which one of the angles is exactly equal to 90o.

The area of the right-angled triangle is

Δ = 1/2 × b × a

where b is the base and a is the height (it is a side perpendicular to the base).

Pythagoras Theorem

Pythagoras theorem also known as pythagorean theorem is a very popular theorem applicable in right angled triangles.

If b is the base, a is the perpendicular and c is the hypotenuse(side perpendicular to the base). Then, according to the Pythagorean theorem:

c2 = a2 + b2

Two types of right angle triangle:

Isosceles Right-angled Triangle: A right-angled triangle in which the remaining two sides have an equal angle of 45o and equal lengths.

Scalene Right-angled Triangle: A right-angled triangle in which the remaining two sides have unequal angles.

Centroid vs orthocenter 

There are three possible bases and consequently, three possible heights. These heights intersect at a point known orthocenter of a triangle.

Whereas the centroid is a point where all three medians of a triangle intersect. 

Let us now solve some example problems.

Solved problems 

Example 1

Consider an isosceles triangle whose perimeter is 24 cm and height drawn to the base is 6 cm. Find the area of the isosceles triangle.

Solution

Area of an isosceles triangle

height = h = 6 cm
perimeter = p = 24 cm

Let A, B, and C be the vertices of this triangle, let b be the base and c be the length of the equal sides. Moreover, let AD represents the height.

Perimeter = a + b + c
But, a=c
Perimeter = 2c + b
24 = 2c + b
b = 24 – 2c → equation 1

Consider the triangle Δ ADB. It is a right-angled triangle.

Let b/2 be the base, c be the hypotenuse and h be the perpendicular. Using Pythagorean theorem

c2 = (b/2)2 + h2

c2 = [(24 – 2c) / 2]2 + 62

c2 = (12 – c)2 + 36

c2 = 144 – 24c + c2 +36

c2 – c= 180 – 24c

24c = 180

c = 180 / 24

c = 7.5 cm

Putting the value of c in equation 1. We get,

b = 24 – 2(7.5)

b = 24 – 15

b = 9 cm

Now, we have the base and the height. We can plug these values into the formula of the area.

Δ = 1/2 × base × height

= 1/2 × 9 × 6

= 27 cm2

The area of the isosceles triangle with the base of 9cm and a height of 6 cm is 27 cm2.

Example 2

ΔABC and ΔDBA are right-angled triangles where ∠BCA = 50o. Find the angles ∠DBC and ∠BAC.

how to find the angle of a triangleSolution

Consider triangle ΔDBC.

∠DBC + ∠BCD + ∠BDC = 180o

Here, ∠BCD = ∠BCA = 50o

∠DBC + 50o + 90o = 180o

∠DBC = 40o

Consider triangle ΔABC.

sum of all angles = 180o

∠BCA + ∠BAC + ∠ABC = 180o

50o + ∠BAC + 90o = 180o

∠BAC = 40o