Proof of Pythagorean Theorem

History of Pythagoras Theorem

Pythagoras theorem is also called the Pythagorean theorem.

Greek Mathematician Pythagoras of Samos was introduced the Pythagoras theorem. He was an ancient Ionian Greek philosopher. He formed a group of mathematicians who works religiously on numbers. Greek Mathematician stated the theorem hence it was named after him as the “Pythagoras theorem.” 

Pythagoras introduced and popularised the theorem, there is sufficient evidence proving its existence in other civilizations, 1000 years before Pythagoras was born. 

What is the Pythagoras Theorem?

If a triangle is right-angled then Pythagoras’ theorem states, that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In triangle ABC, we have BC2 = AB2 + AC2​​. Here, ​​​​AB is the base, AC is the height, and BC is the hypotenuse. Noticed that the hypotenuse is the longest side of a right-angled triangle.

Pythagoras Theorem Formula

Pythagoras Theorem Formula is:


‘c’  is hypotenous.

‘b’ and ‘a’ is the length of the other two sides.

Proof of Pythagorean Theorem using Algebra

A triangle △ABC in which  ∠ABC=90° .

To prove: {(AC)}^2={(AB)}^2+{(BC)}^2 .


Draw perpendicular BO on AC


In △ AOB and △ABC.

∠A = ∠A . common

∠AOB = ∠ABC. AA similarity

\Rightarrow\;\;\;\;\;AO/AB\;\;\;\;=\;\;\;\;\;\;\;AB/AC\; .

\Rightarrow\;\;\;\;AO\;\times\;AC\;\;\;\;=\;\;\;\;\;\;AB^2\; . (1)

\bigtriangleup BOC\;\;\;and\;\;\bigtriangleup ABC.

\angle C=\angle C. common

\angle BOC=\angle ABC\;\;\;equal\;to\;each\;90°.

Therefore \bigtriangleup BOC\sim\bigtriangleup ABC. By AA similarity

\Rightarrow OC/BC\;\;\;=BC/AC.

\Rightarrow OC\times AC\;=BC^2 . (2)

AO\times AC+OC\times AC=(AB^2+BC^2) .

\Rightarrow(AO+OC)\times AC=(AB^2+BC^2) .

\Rightarrow AC\times AC=(AB^2+BC^2) .

\Rightarrow AC^2=(AB^2+BC^2) .

Applications of Pythagoras Theorem

Many applications of Pythagoras theorem can be seen in our daily life. Here, we discussed some applications:

Engineering and Construction fields

The techniques of Pythagoras theorem are used by many architects. If length and width are known, diameter is very easily used.


It is used by sea travelers to find the shortest route and distance to their destination.

Woodwork and interior designing

The Pythagoras concept is used in interior design and the architecture of houses and buildings.

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Azhar Ali

Azhar Ali

Mathematician and Blogger.

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