Question

A figure shows a square made from four right-angled triangles that all have the same dimensions. Each of the four right angled triangles have a height labeled a, a base labeled b, and a hypotenuse labeled c. Sides a and b are positioned so that the right angle creates the four outer corners of the outer square. Each vertex of the inner square divides each side of the outer square in two unequal parts labeled a and b, where a is the shorter part and b is the longer part. Each side of the inner square, labeled c, becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square. The four right-angled triangles are shaded.
Explain a proof of the Pythagorean Theorem using the image.

Answer 1

Place the four identical right triangles inside a larger square of side length a + b so that each triangle’s legs (lengths a and b) lie along the sides of the large square and their hypotenuses (length c) form the sides of a smaller, central square (this is exactly the picture).

Now compare areas of the large square in two ways.

1) As a single square:
area = (a + b)^2.

2) As the union of four triangles plus the central square:
area = 4 × (1/2 ab) + c^2 = 2ab + c^2.

Equate the two expressions:
(a + b)^2 = 2ab + c^2.

Expand the left-hand side and cancel 2ab:
a^2 + 2ab + b^2 = 2ab + c^2
⇒ a^2 + b^2 = c^2.

This is the Pythagorean theorem. The picture provides the geometric arrangement that makes the two area counts obvious.