The Pythagorean Theorem is all about right triangles. It is one of the most popular theorems of mathematics with the help of which we can find the lengths of the sides of a right triangle.
What is a theorem?
Before we discuss about the Pythagorean Theorem, we must know what a Theorem is.
A Theorem is a conjecture or a proposition that has been proved on the basis of those statements which have been established previously.
What is the Pythagorean Theorem?
According to the Pythagorean Theorem, in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
To put it simply,
Where a and b are the lengths of the two legs of a right triangle and c is the length of the hypotenuse.
The Pythagorean Theorem is named after the Greek mathematician Pythagoras, who was born in Samos (ca. 569 – 475 B.C.E). Samos is located on an island of the same name in the Aegean Sea. This ancient town was once the principle commercial center of Greece which now lies in ruins.
We know very little about the life of Pythagoras. None of his writings exist today.
He founded a mathematical society in Croton, now in Italy. The members of that society have discovered irrational numbers as well as the five regular solids. They have also proved the Pythagorean Theorem.
It is interesting to note that although Pythagoras is credited with the discovery of the theorem, it is often argued that the knowledge of the theorem existed before him, as the Babylonian mathematicians have used the formula in their calculations, although there is little evidence that they fitted it into a mathematical framework.
Terms related to the theorem
Right triangle – A triangle whose one of the three angles is a right angle (90 degrees).
Hypotenuse– The side opposite the right angle, in a right triangle, is called the hypotenuse.
Legs– The two sides of a right triangle other than the hypotenuse, is the legs of the triangle.
The diagram above shows a right triangle ABC. AB measures ‘a’ units, BC measures ‘b’ units and AD measures ‘c’ units. AD is the hypotenuse of the triangle since it lies to the opposite side of the right angle ABC, while AB and BC are the two legs.
It is to be Noted:
- The Hypotenuse is the longest side of a right triangle, but not the longest leg.
- The Pythagorean Theorem is not true for any other triangle other than a right triangle.
Did you know?
The Pythagorean Theorem has more than 200 mathematical proofs (derivations) which is probably the most of any mathematical theorem. They include both geometric and algebraic proofs. Elisha Scott Loomis’s Pythagorean Proposition, which was first published in 1927, contains original proofs by Pythagoras, Euclid, Leonardo da Vinci and the U.S. President James Garfield.
One very famous proof of the theorem is done by the Hindu mathematician Bhaskara, one of the first to understand the number system and how to solve equations, much before the Europeans. His proof of the theorem is often called the “Behold” proof since after drawing the diagram Bhaskara said no more than “Behold”.
Converse of the Pythagorean Theorem
So far we have seen that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
The converse of the theorem states that if the three sides of a triangle satisfy the Pythagorean equation, then the triangle is a right triangle.
Three positive integers, for e.g. a, b and c, which satisfy the Pythagorean equation are called Pythagorean triples.
For example,(3 – 4 – 5) is a Pythagorean triple since the sum of the squares of 3 and 4 is equal to the square of 5, i.e., 9 + 16 = 25.
Given below is a list of some Pythagorean triples:(6 – 8 – 10)
(9 – 12 – 15)
(5 – 12 – 13)
(12 – 16 – 28)
The Pythagorean Theorem is used extensively in mathematics, engineering etc. We also find references of this theorem in many books, music, movies etc.