TheoremProved

Pythagorean theorem

Statement

In any right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$ , $a^2 + b^2 = c^2$ . Conversely, if the side lengths of a triangle satisfy $a^2 + b^2 = c^2$ , then the angle opposite the side of length $c$ is a right angle.

Why is it true?

Build a square on each of the three sides of the right triangle: their areas are $a^2$ , $b^2$ , and $c^2$ . If you arrange four copies of the right triangle inside a larger square of side $a + b$ , the uncovered space can be grouped either as one tilted square of area $c^2$ or, after sliding the four triangles into two rectangles, as two smaller squares of areas $a^2$ and $b^2$ . Since the total area and the four triangles never change, the uncovered areas must match: $a^2 + b^2 = c^2$ .

Proof sketch

Drop the altitude from the right angle to the hypotenuse of length $c$ , splitting it into segments of lengths $p$ and $q$ with $p + q = c$ . Each smaller right triangle shares an acute angle with the original triangle, so all three triangles are similar. Comparing ratios of legs to hypotenuses gives $\dfrac{a}{c} = \dfrac{p}{a}$ and $\dfrac{b}{c} = \dfrac{q}{b}$ , or $a^2 = cp$ and $b^2 = cq$ . Adding the two equations yields $a^2 + b^2 = c(p + q) = c^2$ .

Proved by

proved -300Euclid of Alexandria proved -300Liu Hui

Topics that use this theorem

Thales' theorem and the Pythagorean theorem Trigonometry Ancient Mathematics

Related theorems

Law of cosines Thales's intercept theorem Heron's formula Ptolemy's theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Euclid (translated by Thomas L. Heath) (1956). The Thirteen Books of Euclid's Elements, Vol. 1 (Book I, Propositions 47–48)
Eli Maor (2007). The Pythagorean Theorem: A 4,000-Year History