Pythagorean Proof

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Pythagorean Proof

  1. 1. Proof For Pythagorean Theorem Jigar Bhatt 9769907927
  2. 2. Pythagorean Theorem <ul><li>Pythagorean theorem states that, “The sum of the squares of </li></ul><ul><li>the sides of a right triangle other than hypotenuse is equal to </li></ul><ul><li>the square of hypotenuse.” </li></ul><ul><li>c 2 = a 2 + b 2 </li></ul>A C B b a c
  3. 3. Proof <ul><li>Consider a right triangle ABC. In the triangle AB= c, AC= b, </li></ul><ul><li>BC= a, </li></ul><ul><li>Now we have to prove that, c 2 = a 2 + b 2 </li></ul>A C B b a c
  4. 4. <ul><li>Now extend line CB to point X 1 such that BX 1 = b, </li></ul><ul><li>Extend line CA to point X 2 such that BX 2 = a. </li></ul>Proof A C B b a c X 1 X 2
  5. 5. Proof A C B b a c X 1 X 2 K 1 K 2 Now draw a line X 1 K 1 such that X 1 K 1 = b and X 1 K 1 is perpendicular to CX 1 . Draw a line X 2 K 2 such that X 2 K 2 = a, and X 2 K 2 is perpendicular to CX 2 . b a b a
  6. 6. Proof A C B b a c X 1 X 2 K 1 K 2 Extend a line X 1 K 1 to point X 3 such that K 1 X 3 = a and X 1 K 1. to CX 1 . Extend a line X 2 K 2 to point X 3 . b a b a a X 3
  7. 7. Proof A C B b a c X 1 X 2 K 1 K 2 Constructing this lines we have made a square whose side is (a+b). If we observe carefully we have four right triangle at each corner of square. b a b a a X 3 b
  8. 8. Proof A C B b a c X 1 X 2 K 1 K 2 We have four congruent right triangles. A square with side c, and a square with side (a+b). Area of square with side (a+b) = 4 x Area of right triangle + Area of square with side c b a b a a X 3 c c c
  9. 9. Proof Area of square with side (a+b) = 4 x Area of right triangle + Area of square with side c (a+b) 2 = 4 x (1/2) a x b + c 2 a2 + 2ab + b2 = 2ab + c 2 - 2ab - 2ab a 2 + b 2 = c 2 Hence Proved.
  10. 10. Thank you

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