Pythagoras’ theorem 1

This is a famous theorem and has many interesting proofs. It is known as Pythagoras’ theorem but it was known by the Greeks, Chinese, Indians, Arabs, Persians way before the Greek mathematician Pythagoras’ birth but it is his name which is carried by this theorem.

Wait, we learnt about notions, axioms, and postulates, now what is a theorem and proof? Let us figure out a few of these terms before proceeding with Pythagoras’ theorem also known as Pythagorean theorem.

Definition: A concise and precise explanation of a mathematical word.

Theorem: An important true statement.

Proof: An explanation of why a statement is true or  false.

Lemma: A true statement used to prove another statement to be true.

Axiom: A basic assumption about a mathematical statement.

Converse: To switch the hypothesis with the conclusion.

And now back to ‘an important true statement’, known as the Pythagorean theorem.

Pythagorean theorem

In a right triangle, the sum of the squares of the legs (a & b) of the right angle is equal to the square of the hypotenuse (c).

a^2 + b^2= c^2

The  Converse of Pythagorean theorem:

If the sum of the squares of the legs (a & b) of the right angle is equal to the square of the hypotenuse (c)in a triangle, then the triangle is a right angled triangle.

So the converse of Pythagorean theorem is also true.

A corollary to the converse of Pythagorean theorem:

Let c be chosen to be the longest of the three sides and a + b > c (otherwise there is no triangle according to the triangle inequality). The following statements apply:

  • If a2 + b2 = c2, then the triangle is right.
  • If a2 + b2 > c2, then the triangle is acute.
  • If a2 + b2 < c2, then the triangle is obtuse.

 

Uses of Pythagorean theorem:

1> To find the length of hypotenuse of a right angled triangle when the length of the other two sides are known.

2> To find the length of any side of a right angled triangle when the length of the other two sides are known.

3> To find out if a triangle is a right angled triangle,acute or obtuse.

Apart from these uses this theorem has many uses in higher level maths, architecture, physics etc.

 

 

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