The Pythagorean Theorem — Visual Proofs

What just happened?

The square built on the hypotenuse has exactly the same area as the sum of the squares built on both legs. You can see it without a formula: $9$ blue and $16$ green tiles, once rearranged, fill $25$ cells of the square on the hypotenuse — with no gaps and no overlap.

The Pythagorean theorem:

In a right triangle with legs $a$, $b$ and hypotenuse $c$ we have:

For the $3\text{–}4\text{–}5$ triangle this gives $3^2 + 4^2 = 5^2$, that is $9 + 16 = 25$. It is one of the oldest and most important theorems in geometry — attributed to Pythagoras and the Pythagoreans (c. 500 BC), though it was known earlier to the Babylonians and in India.

Proofs "without words":

The theorem has hundreds of different proofs — many of them elegant visual proofs, based on rearranging and laying out the same figures. The same set of pieces, arranged in two ways, forms once $a^2+b^2$, and once $c^2$ — and since they are the same pieces, their areas are equal.

The invariant — the right-angle condition:

The equality $a^2+b^2=c^2$ holds only for a right triangle (a right angle between the legs $a$ and $b$). For an acute angle $a^2+b^2>c^2$, for an obtuse angle $a^2+b^2<c^2$. The converse says that if $a^2+b^2=c^2$, then the triangle must be right-angled — that is how a right angle is recognised.

Why you need this for the exam:

The Pythagorean theorem is the foundation of Euclidean geometry, trigonometry and analytic geometry — the formula for the distance between points $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ is Pythagoras directly. Trick: recognise the right angle, label the legs and the hypotenuse, substitute into $a^2+b^2=c^2$.

Remember: in a right triangle, the area of the square on the hypotenuse is the sum of the areas of the squares on the legs — $a^2+b^2=c^2$ (e.g. $3^2+4^2=5^2$).