Posted by: dreamofdestiny | Wednesday, 05 Nov 2008

Pythagoras’ Theorem

Every math students know the Pythagoras’ Theorem, largely identified as:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

or quite simply:

The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.

Graphically represented as follow:

Pythagoras' Theorem
Pythagoras’ Theorem

Rarely a student is recquired to proof the equation above. Mostly in assignments, they are recquired to apply this equation to solve other mathematical problems. So if I were to ask you to prove the equation above, how would you go about doing it?

There are many ways to prove the Pythagoras’ Theorem, through graphically, algebraically, and even through calculus. Some are simpler to understand than others. Since I am not running a fully mathematical blog, I am going to stick with graphics plus some simple algebraic equations, which should be easy to understood.

Consider the following diagrams and the text that accompanies them:

Full Square
Diagram A
Diagram B
Diagram B
Diagram C
Diagram C

The explanation that follows are equations based on the diagrams presented above.

They should provide proof of the Pythagoras’ Theorem graphically.

In Diagram A, the following is true:

Area (A1) = (a + b)2

In Digram B, the following is observed:

Area (A2) = c²

In Diagram C, the following is seen:

Area (A3) = (a x b) / 2

Now, if you see from the three diagrams above, it can be deduced that:

4 of Diagram C plus 1 of Diagram B equals to 1 of Diagram A

In equation format, this can represented and clarified as below:

4 x [(a x b) / 2] + c² = (a + b)²

2ab + c² = a² + 2ab + b²

c² = a² + b²

So, there you have it. The proof of Pythagoras’ Theorem.

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Responses

  1. If I know the hypotenuse to be 20 inches what is formula that would tell me the must different combinations on the other (2) sides. They have to be whole numbers.

    Thanks.

  2. thanks james. i would explain this in my next post in mathematics area 🙂

  3. […] Triple, Part 0 Quite recently, I received this comment on my Pythagoras’ Theorem post: If I know the hypotenuse to be 20 inches what is formula that would tell me the must […]

  4. Hi. I have also created a proof of the Pythagorean theorem here:

    http://math4allages.wordpress.com/2010/02/03/pythagorean-theorem/

    You may want to check it out.


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