University:
California State University, Northridge
Course:
MATH 310 | Basic Concepts of Geometry
Academic year:

2008
Views:

475

Pages:

1
Author:

Aditya Porter

A Proof of the Pythagorean Theorem There are hundreds of proofs of the Pythagorean Theorem. The following proof is attributed to the Indian astronomer Bhaskara (1114-1185). b a 1 a Given any right triangle with sides a,b,c 3 2 c b We can construct a square with sides of length a + b, so four copies of the given right triangle can be placed as shown, forming a c-by-c square inside the a+b square. b c c b c a c a That the c-c-c-c rhombus in the center is a b indeed a square can be verified by observing that the angles marked 1 and 2 are complementary (since they are the acute angles of a right triangle, copied four times), and that 1 & 2 & 3 form a straight (180E) angle. (If you don’t see this, try the following: mark all the angles that are congruent to angle 1 with a “1", and all angles that are congruent to angle 2 with a “2" . Also keep in mind that the triangle is a right triangle.) So, now that we know for sure that the square in the center truly is a square.... We argue that the area of the large square must equal the sum of the areas of the smaller square and the four right triangles: The area of the large (a+b) square is (a+b)(a+b) = a2 + 2ab + b2 . The area of the smaller square plus the four triangular areas = cCc + 4C(½)ab = c2 + 2ab . Since these must be equal: a2 + 2ab + b2 = c2 + 2ab a2 c2 Therefore: + b2 = — QED ( Quod Erat Demonstratum.) By the way, the illustration of the a+b-sided square above appears “cockeyed”; that is an optical illusion caused by the interior c-by-c square.