 #### Pythagoras theorem

Many amazing discoveries are made in the field of mathematics. One such amazing discovery about the triangles is the Pythagoras theorem. It is used in various fields such as navigation and construction.

Pythagoras theorem is also known as Pythagorean theorem. It gives the relationship between the three sides of a right angled triangle.

PQR is a triangle with one of its angles as 90 oC. The side opposite to the right angle, i.e. PR is the hypotenuse.

The Pythagoras theorem says that the hypotenuse has a special relationship with the other two sides of the triangle.

In the triangle PQR,

PQ = a, QR = b, PR = c

According to Pythagoras theorem,

(Hypotenuse)2 = Sum of the squares of the other two sides i.e. c2 = a2+b2

How to derive Pythagoras theorem?

The process to understand Pythagoras theorem can easily be explained using an equation i.e. (a+b)2.

Draw a line of any length and mark a point on it. Name the segments as ‘a’ and ‘b’.

Length of the line = a+b

Let us complete the square with (a+b) as its side and make the same division as in the line.

Now, there are four points one on each side of the square.

Join the adjacent points as shown in the figure. The figure formed by joining the points is a square.

Let the length of the side of the new square be ‘c’ units.

The outer square is made up of five parts, i.e. a smaller square of side ‘c’ unit and four right angled triangles with height ‘a’ and base ‘b’.

Area of outer square = Side x Side
= (a+b) x (a+b)
= (a+b)2  ———————-(1)

Again, area of outer square = Sum of area of 5 parts
= (Area of small square) + 4(Area of triangle)
= c2 + 4 (½ ab)
= c2 + 2ab ———————-(2)

Equating equations (1) and (2)

(a+b)2 = c2 + 2ab
c2 + 2ab = (a+b)2
c2 + 2ab = a2 + b2 + 2ab
c2 = a2 + b2

Thus, if we have a right angled triangle with height ‘a’, base ‘b’, and hypotenuse ‘c’,

i.e. c2 = a2 + b2  (by Pythagoras theorem)

Hence, proved.

Example of Pythagoras theorem

Pythagoras theorem has innumerable uses in daily life like it is used in architecture, woodworking, or other physical construction projects.
Let us take a simple example.

Kapil wants to build a ramp from the ground leading to the step. The step is 3 feet off the ground and according to building regulation, the ramp must start 8 feet away from the base of the step. How long will the ramp be?

By, Pythagoras theorem,

x2 = 32 + 82
x2 = 9 + 64
x = √73
x = 8.54 ft

Thus, now Kapil knows how long the ramp should be. 