The page has 64 proofs of the Theorem. Is yours proof number 4?
okay, ive got a big square, with a square inside it diagonally. leaving 4 right angled triangles. one at each cornor of the large square.
sides of the internal squares are 'z' and the 3 sides of triangle are x y and z.. z is the hypotenuse...
i have shown that the large square's area can be written as (x+y) squared and also as z squared + 2xy.
now i have to use these results to prove pythagoras theorum?!?!
can any1 help with that? like now.. lol i have to give it in 2moro at skool.
Ummm, you were able to show that the area, A, of the large square is:Originally Posted by adam gerrett
A = (x+y)^2 -----------(1)
Or,
A = z^2 +2xy -------------(2) ?
Very good. In fact, those two equations are both correct.
Each side of the large square is (x+y), so, A = (side)^2 = (x+y)^2.
Each of the 4 right triangles has an area of (1/2)(x)(y) = (1/2)xy,
so the 4 rigth triangles have a collective area of 4*(1/2)xy = 2xy.
The area of the inner, smaller square of side z is z^2, hence the areaof the large square is
A = 2xy +z^2
or, A = z^2 +2xy.
Now, since A = A, then,
(x+y)^2 = z^2 +2xy
Simplifying,
x^2 +2xy +y^2 = z^2 +2xy
The 2xy cancells out,
x^2 +y^2 = z^2 ----------on any of the 4 right tgriangle. The Pythagorean Theorem!
On any of the 4 right triangles,
---z is the hypotenuse
---x and y are the two legs
z^2 = x^2 +y^2
Phythagorean Theorem.
First he is going to really mad.Originally Posted by CaptainBlack
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I made up my own proof to the pythagorean theorem using circles maybe I will post it. I am just warning you that I might not because it is difficult to draw geometrical shapes through the internet.
As promised, here is my proof.
I searched the 69 proofs and did not seem to be there.
2)Triangle ABC with right angle at C.
3)Using AC draw a circle with that radius at C.
5)The circle in 2) intersect hypotenuse AB at D because it is longer than the radius.
7)Extend AD so that it intersect this circle at E.
11)EB is a secant by definition.
13)BC is a tangent at C because C is a right angle.
17)Thus, .
19)Thus,
23)Thus,
29)Thus,
31)Thus,
37)Thus,
39)Thus,
41)Thus,
43)Thus,
Notes,
7)Postulate #2
13)A line perpendicular to the radius on the circumfurence is the tangent.
17)The theorem of tangent and secants.
19)Because since it is a line.
23) since it is radius of circle.
29)Add them.
31)Distributive property of the real numbers.
37)Addition is well-defined.
39)A complete square, again distributive.
41)Radius of circle.
43)Addition of line segments.
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Tell me if you need a picture I think it is good enough.