Pythagorean Theorem proof : Geometry / Trigonometry

Pythagorean Theorem proof : Geometry / Trigonometry


Today we are going to discuss on Pythagorean theorem. Pythagoras was a Greek philosopher and mathematician and he had key interest in metaphysics, music and politics also. One of his greatest inventions is Pythagorean theorem. Before we prove this theorem, let us see what “Pythagorean theorem” is So take a right angled triangle and here is the right angle This is the right angle and three sides of this triangle are ‘a’, ‘b’ and ‘c’. So here ‘a’ is the perpendicular of this triangle, ‘b’ is the base and ‘c’ is the hypotenuse. So, in this situation, here, ‘c’ squared is equal to ‘a’ squared added with ‘b’ squared. This is Pythagorean theorem. So to prove Pythagorean theorem we should have a rectangle whose each side is of length a + b So, here, this side is a + b. we split this side into two parts one is ‘b’ and the other one
is ‘a’ similarly we split this side also into two parts ‘b’ and ‘a’ Similarly here also. ‘b’ and ‘a’ and again here also ‘b’ and ‘a’ Now if we join these points We will end up with another square So, that square is this one. and each side of this small square – let us assume that is ‘c’ so we end up with square of side ‘c’ and four right-angled triangles, the sides of each triangle are ‘a’, ‘b’ and ‘c’ Look __ these yellow triangles So each and every triangle here is having area half into perpendicular into base. now if we evaluate the area of the bigger square as each side is of length a + b the area is the side squared So that is equal to ‘a’ squared added with ‘b’ squared added
with twice of ‘a’ times ‘b’ And now area of four triangles is equal to four times the area of one triangle So that is equal to twice of ‘ab’ Now from the figure it is clear that the area of smaller square is the area of bigger square subtracted by area of four triangles that means the area of smaller sqare is this one which is the area of bigger square
attracted by four triangles a squire subtracted by area of four triangles a squire So ‘a’ squared added with ‘b’ squared added with 2 x a x b subtracted by 2 x a x b so that should be equal to ‘a’ squared plus ‘b’ squared and we know that the area of smaller square as this is square having each and every side is equal to ‘c’ So the area of this smaller square should be equal to ‘c’ squared and that is equal to ‘a’ squared added with ‘b’ squared And this is Pythagorean theorem