Prove the Pythagorean Identity using a diagram? And also prove that Area of triangle = a^s sinC x sinB/ 2 Sin(B+C)?

You do not need to go to TRIGONOMETRY to prove this.
Simple arithmetic and algebra should do it.

a^2 + b^2 = c^2

(3*3) + (4*4) = (5*5)

9 + 16 = 25.

If you must do it geometrically, try using President Garfield's trapezoid method of proving the Pythagorean theorem.

A mathematical proof requires starting with information which is known and accepted as truth and using only these facts, showing through a series of steps that the hypothesis is true. Picking numbers that work in the hypothesis equation to show one example of a calculation is not a mathematical proof. In fact, most mathematical proofs never are carried into a numerical solution since there would be an infinite number of numerical examples that are true. My old math prof, years ago said something to the effect of, "I don't care if you give me 100,000 examples that are true; that does not PROVE that someone else may find one example where it is not true."

The Garfield graphical solution suggested by Galactus is good and a recognized proof of the theorum. One thing that has always bugged me a little about it is that it requires that the formula for the area of a trapezoid is already in your start bag of truths.

For any student asking for help with proofs, it would be good to list the applicable knowns after the definition of the problem. This used to be required as part of the solution. If done properly it can lead you towards the solution. And, for others to help you, it provides a definition of your starting truths which can vary in the classroom based on what has previously been proven.

Zazonker,

In fact, most mathematical proofs never are carried into a numerical solution since there would be an infinite number of numerical examples that are true.

The more examples that are true, the BETTER.

The 3,4 and 5 combination is very well known.
Do you know of any others ?

Just Google it.

Check this out.

Pythagorean Theorem

galactus,

Do you know of any others ?

Please answer the question.
It is of course obvious that any multiples
Of 3,4 and 5 would qualify.

Quote:

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Originally Posted by ROLCAM

galactus,

Do you know of any others ?

Please answer the question.
It is of course obvious that any multiples
of 3,4 and 5 would qualify.

---

Do you mean are there more Pythagorean triples?

Of course, many more. Here's some:

(20, 21, 29), (11, 60, 61), (13, 84, 85), (5, 12, 13) , (12, 35, 37) , (16, 63, 65) ,(36, 77, 85)
(8, 15, 17) ,(9, 40, 41) ,(33, 56, 65) ,(39, 80, 89),(7, 24, 25) ,(28, 45, 53) ,(48, 55, 73) ,
(65, 72, 97)

81 approaches to proving the Pythagorean theorem:

Pythagorean Theorem and its many proofs from Interactive Mathematics Miscellany and Puzzles

Rolcam. It's not about calculating at all. It's not arithmetic or algebra. Your first example, for example, "showed" that 3*3 + 4 *4 = 5*5. That is arithmetically true. But, it doesn't PROVE that three connected lines of lengths 3, 4 and 5 would even meet at all end points to form a triangle. The mere fact that we all know that they will, does not provide mathematical proof. -- I'm not trying to fight a war with you. I'm just concerned that you while you don't understand the concept, you are making very strong inapplicable statements. This under the heading of "Expert" can only confuse people who are struggling with understanding what a proof is.

Here's an excerpt from Wikipedia. Please read it carefully. Notice that plugging numbers into the equation to get a bunch of examples is not part of the process.

In mathematics, a proof is a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproved proposition that is believed to be true is known as a conjecture.

The statement that is proved is often called a theorem. Once a theorem is proved, it can be used as the basis to prove further statements.

Why would give me a disagree when I was trying to help? I am not sure what you are after. There are proofs of the Pythagorean theorem all over the internet.

Here is a proof I always liked using Differential Equations. I just remembered this one from way back. It can most likely be found on the web as well.

By noting how the change is a side changes the hypoteneuse.



Separate variables:



Integrate:



If a=0, then b=c.

So the constant is