# !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!*

## 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:

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

## ؟؟How to find the hypotenuse c?? By using the pythagorean theorem: C^2 =a^2 +b^2 C^2 = 200^2 +210^2 C^2=40000+44100 C^2= 84100 C=290 | ## ؟؟How to find the measure of the ground??Let ladder be C Let ground be A Let wall be B By using the pythagorean theorem: C^2=a^2+b^2 13^2=a^2+12^2 169=a^2+144 A^2=169-144 a^2=25 A=5
| ## ؟؟How to find the measure of h?? By using the pythagorean theorem: C^2=a^2+b^2 10^2=5^2+b^2 100=25+ b^2 B^2= 100-25 B^2= 75 B= radical 75 |

## ؟؟How to find the hypotenuse c??

C^2 =a^2 +b^2

C^2 = 200^2 +210^2

C^2=40000+44100

C^2= 84100

C=290

## ؟؟How to find the measure of the ground??

Let ladder be C

Let ground be A

Let wall be B

By using the pythagorean theorem:

C^2=a^2+b^2

13^2=a^2+12^2

169=a^2+144

A^2=169-144

a^2=25

A=5

*Note: we choose the positive square root only since we can't measure lengths in negative numbers*

## !!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:**

## 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**

*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!*