Simple Proofs of Pythagorean Theorem

Explore some simple proofs of the Pythagoran theorem.

Proof 1

In the figure below are shown two squares whose sides are a + b and c. let us write that the area of the large square is the area of the small square plus the total area of all 4 congruent right triangles in the corners of the large square.

Pythagora theorem proof 1


(a + b)
2 = c2 + 4 (1/2) (a*b)

Expand the left hand side of the above equality, and simplify the last term on the right

a
2 + b2 + 2 a*b = c2 + 2 a*b

Simplify to obtain

a2 + b2 = c2
a and b are the sides of the right triangle and c is its hypotenuse.


Proof 2

We first start with a right triangle and then complete it to make a rectangle as shown in the figure below which in turn in made up of three triangles.

Pythagora theorem proof 2


The fact that the sides of the rectangle are parallel, that gives rise to angles being congruent (equal in size) in all three triangles shown in the figure which leads to the triangles being similar. We first consider triangles ABE and AED which are similar because of the equality of angles. The proportionality of corresponding sides (a in triangle ABE corresponds to c in triangle AED since both faces a right angle, x in triangle ABE corresponds to a in triangle AED both faces congruent (equal) angles) gives

a / c = x / a which may be written as a
2 = c*x

We next consider triangles ECD and AED which are similar because of the equality of angles. The corresponding sides are proportional, hence

b / c = y / b which may be written as b
2 = c*y

We now add the sides of the equalities a
2 = c*x and b2 = c*y to obtain

a
2 + b2 = c*x + c*y = c(x + y)

Use the fact that c = x + y to write

a
2 + b2 = c2



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