proof of pythagoras theorem?????

pls help me out

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  • 1 decade ago
    Favorite Answer

    It is easy to grasp and prove Pythogorous theorem by relating right angle triangles having "whole number sides lengths alone"!

    3^2+4^2+5^2 is graphically explainable on computers.

    (Rearrange it as 5^2-4^2= 3^2. Note that 3^2=9 and (5+4) = 9

    Further, 5^2 is 25 and by a split of it as 13 and 12 we have (13^2-12^2) = 5^2

    And so on, endless possibilities of (odd number)^2 split and relating to separate right angle triangles exists!

    A circle of "radius 5" has 8 number of " x,y whole number points" (2 per 2D-quadrant) and another four points on x and y axes (where either a-'x' or a-'y' is zero). In all 12 points with "x and y as whole numbers".

    Similarly a circle of "radius-13" too has 8 numbers x,y whole number points (2 per a- quadrants) and another four points on x and y axes (where either a -x or a- y is zero). In all 12 points exist with "x and y as whole numbers".

    Above two triplets actually relates smaller near-zero number sets "3, 4, 5" and "5, 12, 13". And we have

    3^2+4^2= 5^2.................(1) and "5" is a hypotanuse.

    5^2+12^2= 13^2.............(2) 5 and 13 are hypotanuses.

    We have 3 hypotanuses in above two formulae. '5' and '5,13'

    "3, 4 and 12" are not hypotanuses in both formulae!

    Now imagine a brick of size 3 units (height), 4 units (breadth) and 12 units (length). Said brick has volume 3* 4* 12 = 12^2 = 144 cubic units!

    (3^2 +4^2 = 5^2 is not only an equality in between 'sum of square units of two sides' and 'a hypotanuse square unit' but also extends as a volume and surface area relations as stated herinafter!

    We know that 12^2 cubic units is volume of brick!

    Surface area of brick is 5*4*12 =240 sq units which is "5 multiplied by base area (48 sq units) /( 4 *12) " Said relations exist when said pair of triplets relate 'a brick size'!

    Probably said relatins were knowingly fixed by first users of brick having 3 inches height, 4 anches wide and 12 inches long sizes! It reveals an ancient manner of proving utility of pythagorus theorem!

    You may regard above facts as a practical application of pythogorus theorem. It also give an insight into "area and volume" relations, which has consistently helped users of theorem!

    Regards!

  • 1 decade ago

    Draw a right triangle ABC.Draw a perpendicular to the hypotenuse from the opposite vertex and then proof the theorem by similarity.

    I can't give u the proof as I can't sketch the diagram but u can find it in any 10th standard book.

  • Anonymous
    1 decade ago

    Hi Annie, this is Kekin, ur new friend, here is ur answer,

    This theorem may have more known proofs than any other (the law of quadratic reciprocity being also a contender for that distinction); the book Pythagorean Proposition, by Elisha Scott Loomis, contains 367 proofs.

    & search this site

    http://en.wikipedia.org/wiki/Pythagoras_theorem

    Thanks,

    bye!

    Kekin

    Source(s): kekin_patel@yahoo.com
  • 1 decade ago

    Check this link. It has a very comprehensive list of proofs.

    http://www.cut-the-knot.org/pythagoras/index.shtml

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  • 4 years ago

    Draw a square ABCD where each side length a + b units long. Draw point X on AB so that AX = a and BX = b. Draw point Y on BC so that BY = a and CY = b. Draw point Z on CD so that CZ = a and DZ = b. Draw point W on AD so that DW = a and AW = b. Now connect points XYZW. Since <A = <B = <C = <D = 90, note that by SAS congruence theorem that triangles: WAX, XBY, YCZ, ZDW are congruent to each other. Thus, corresponding sides and angles are congruent. Then XY = YZ = ZW = WX. Thus, all side lengths are equal. Let c be this common side lenght. Also, <AWX = <BXY = <CYZ = <DZW and <AXW = <BYX = <CZY = <DWZ. Note that <AXW + <AWX + <WAX = 180 since the sum of angles in a triangle is 180. Then <AXW + <AWX + 90 = 180 since <WAX is a right angle. Then <AXW + <AWX = 90. Note that <AXW + <WXY + <BXY = 180 since A,X,B are collinear. Then <AXW + <WXY + <AWX = 180 since <BXY = <AWX. Then <WXY + 90 = 180 since <AXW + <AWX = 90. Then <WXY = 90. Thus, <WXY is a right angle. Similarly, <XYZ = <YZW = <ZWX = 90. Thus, all angles are equal to 90. Now since all sides are equal and all angles are equal we see that XYZW is a square with side length c. Let's compare the areas. Area of square ABCD is (a + b)^2. Now look at it as a sum of the four triangles and square XYZW. Then the area is 4 * (1/2)(ab) + c^2. Then (a + b)^2 = 4*(1/2)(ab) + c^2 a^2 + 2ab + b^2 = 2ab + c^2 a^2 + b^2 = c^2 DONE!

  • Anonymous
    1 decade ago

    The most common proof is the geometrical one:

    http://en.wikipedia.org/wiki/Pythagorean_theorem

  • Anonymous
    1 decade ago

    The proof is more geometrical and logical... I cant really portray it with mathematics symbols or equations. It really needs to be done visually.

  • Robin
    Lv 4
    1 decade ago

    there are many ways to prove it

    in what grade are you?

    do you have any givens?

    eg

    ab^2=bc^2+ca^2-2bc*ac*cosc

    but when c=90 deg

    =>pyth. is true

  • Pranil
    Lv 7
    1 decade ago

    See the answer given by Daniel & T.T.K.

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