# Intuition behind Pythagoras Theorem

The Pythagoras Theorem states that in a right angled triangle, ‘a’ being the base, ‘b’ being the height and ‘c’ being the hypotenuse of that triangle, then a^{2}+b^{2}=c^{2}

Below is an illustration of this –

**Example –**

**1. if the base of a right angled triangle is 3, the height is 4,then what is the length of its hypotenuse?****Solution – ** given, a=3, b=4 ,c=?

Using Pythagoras theorem,

a^{2}+b^{2}=c^{2}

3^{2}+4^{2}=c^{2}

√(9+16) =c

c=5

**2. if the hypotenuse of a right angled triangle is 13, the height is 5,then what is the length of its base?****Solution –**

given, a=?, b=5 ,c=13

Using Pythagoras theorem,

a^{2}+b^{2}=c^{2 }

a^{2}+5^{2}=13^{2}

a=√(169-25)

a=12

**Intuition behind Pythagoras Theorem :**

Let’s prove this theorem using the figures.

Draw squares corresponding to each side of the triangle as follows –

If we look at the figure closely, we could reframe the pythagoras theorem as follows-

The area of 2 squares is equal to the third square.

ie- a^{2} is the area of the 1st square

b^{2} is the area of the 2nd square

c^{2} is the area of the 3rd square

thus, a^{2}+b^{2}=c^{2}

Another proof of pythagoras theorem can be shown by rearranging the triangles to form 2 squares as follows

If we compare the two squares, we can find that both the squares have a+b side length, thus having the same area.

In each square, four right-angled triangles are used (realigned in a different way though)

So, we can conclude that

area(1st square) =a rea(2nd square)

c^{2} + 4* (area of a right angled triangle)= a^{2}+b^{2}+4* (area of a right angled triangle)

c^{2}=a^{2}+b^{2} [cancelling the common terms from both sides]

Thus, pythagoras theorem is proved.