!PYTHAGOREAN THEOREM!

Classmates, let me introduce you to the pythagorean theorem!

Who discovered the pythagorean theorem?

Before getting to know the pythagorean theorem , we must know who discovered it and that would be the great mathematician ,philosopher and scientist, PYTHAGORAS.


Pythagoras was born on 570 BC in Samos, Greece. He lived with his wife theano and four children arignote, myia ,damo and telauges.

WHAT IS THE PYTHAGOREAN THEOREM?

Over 2000 years ago there was an amazing discovery about triangles:


When a triangle has a right angle (90°) ...

... and squares are made on each of the three sides, ...... then the biggest square has the exact same area as the other two squares put together!

It is called the PYTHAGOREAN THEOREM

It can be written in a simple equation which is a^2 +b^2 =c^2

Note1: C is the longest side of the triangle and A and B are the two other sides as follows:

Note2: the longest side of a right triangle is called the hypotenuse

THUS AFTER OBSERVING THE PREVIOUS DIAGRAMS AND EXPLANATIONS WE CAN DEDUCE THE DEFINITION OF THE PYTHAGOREAN THEOREM:

Definition:

In a right angled triangle:

the square of the hypotenuse is equal to

the sum of the squares of the other two sides.


To confirm this theorem we are going to perform the following exercise :

Example: A "3,4,5" triangle has a right angle in it.



Let's check if the areas are the same:

3^2 + 4^2= 5^2

Calculating this becomes:

9 + 16 = 25

It works ... like Magic!

Note3:

Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived !

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WHY IS THIS THEOREM USEFUL?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!) and also this isn't its only usage! By using its CONVERSE we can prove right triangles as shown below:

THE CONVERSE OF THE PYTHAGOREAN THEOREM:

The converse of the pythagorean theorem states that: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Example: Does this triangle have a Right Angle?


Does a^2+ b^2= c^2 ?

  • a^2+ b^2= 102 + 242 = 100 + 576 = 676
  • c^2= 26^2 = 676

They are equal, so ...

Yes, it does have a Right Angle!

NOTE4:

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Pythagorean theorem water demo

IN ORDER TO COMPREHEND THE PYTHAGOREAN THEOREM MORE I'LL BE SHOWING YOU SOME REAL LIFE EXAMPLES !

Colin Dodds - Pythagorean Theorem (Math Song)
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!!NO.. WE ARE NOT DONE YET!!

The pythagorean theorem is not only used in right triangles, it is used in special cases of right triangles too which are:

FIRST CASE: A RIGHT ISOSCELES TRIANGLE

The pythagorean theorem in a right isosceels triangle states that:

The hypotenuse of the right isosceles triangle is equal to side*radical 2

As shown in the examples below:

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SECOND CASE: A SEMI EQUILATERAL TRIANGLE

The pythagorean theorem ina semi equilateral triangle states that:

The side facing 30 degrees is equal to hypotenuse/2

The side facing 60 degrees is equal to (hypotenuse *radical 3)/2

As shown in the examples below

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I HOPE THAT YOU ENJOYED MY EXPLANATION AND ENJOYED THE LESSON

AND NOW YOU ARE ABLE TO USE THE PYTHAGOREAN THEOREM, AND PROVE IT YOURSELF TOO!