Pythagorean Theorem

The Pythagorean theorem is a fundamental result in geometry stating that, in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. If the perpendicular sides, or legs, have lengths $a$ and $b$, and the hypotenuse has length $c$, then $a^2+b^2=c^2$. Geometrically, the square constructed on the hypotenuse has the same area as the two squares constructed on the legs together. (aleph0.clarku.edu)

Statement and converse

The theorem concerns triangles in Euclidean geometry. The hypotenuse is the side opposite the right angle, and the letters in the formula must be assigned accordingly. Solving the equation for a missing side gives

The positive square root is used because side lengths are positive. For example, perpendicular sides of lengths 3 and 4 give a hypotenuse of length 5, since $3^2+4^2=25$. The relationship applies to arbitrary positive lengths, not merely whole numbers. (personal.math.ubc.ca)

The converse is also a theorem: if a triangle has side lengths satisfying $a^2+b^2=c^2$, the angle opposite side $c$ is a right angle. Thus the equality both describes right triangles and provides a test for identifying them. In Euclid’s Elements, the theorem and its converse appear as Book I, Propositions 47 and 48. (mathcs.clarku.edu)

Historical development

The name commemorates Pythagoras, a philosopher and mathematician of ancient Greece, but does not establish that he first discovered the relationship. Surviving attributions were written centuries after his lifetime, and no proof by Pythagoras himself survives. It is therefore important to distinguish the theorem’s conventional name from documented evidence of its origins. (aleph0.clarku.edu)

Babylonian mathematical documents from Mesopotamia demonstrate knowledge of numerical relationships associated with right triangles long before Pythagoras. The clay tablet Plimpton 322, dated approximately to 1800 BCE, contains numbers related to Pythagorean triples. Such evidence establishes early numerical knowledge, but should not automatically be equated with a surviving deductive proof of the general theorem. (personal.math.ubc.ca)

Euclid’s surviving presentation treats the theorem through areas and previously established geometric propositions. His Book I proof does not require a general theory of proportional lengths; an alternative treatment using similarity appears later in the Elements. (aleph0.clarku.edu)

Proof methods

Many mathematical proofs are known. Their methods include rearranging pieces without changing area, comparing equal-area figures, and using triangle similarity. These approaches demonstrate the same relationship through different geometric constructions. (personal.math.ubc.ca)

In a rearrangement proof, four identical right triangles with legs $a,b$ and hypotenuse $c$ are placed inside a square of side $a+b$. Their hypotenuses enclose a central square of side $c$. Comparing the outer square’s area with the areas of the four triangles and central square gives

Expanding the left side and cancelling $2ab$ yields $a^2+b^2=c^2$. The construction’s justification includes showing that the central figure really has four right angles. (personal.math.ubc.ca)

A similarity proof draws the altitude from the right-angle vertex to the hypotenuse. If the altitude divides the hypotenuse into segments $p$ and $q$, adjacent respectively to legs $a$ and $b$, similarity gives

Adding these equations and using $p+q=c$ proves the theorem. (aleph0.clarku.edu)

Integer solutions and irrational lengths

A Pythagorean triple consists of three positive integers satisfying the equation. Examples include $(3,4,5)$ and $(5,12,13)$. Multiplying every member by the same positive integer produces another triple. A triple is primitive when its members have no common factor greater than 1. (personal.math.ubc.ca)

A standard parametrization in number theory is

where $m>n>0$ are integers. These expressions satisfy the equation by expansion. When $m,n$ are coprime and of opposite parity, the resulting triple is primitive; every primitive triple arises this way, allowing interchange of the legs. (personal.math.ubc.ca)

Not all right triangles have rational side ratios. A square with side length 1 has diagonal length $\sqrt{2}$, an irrational number. This illustrates the distinction between exact geometric lengths and lengths expressible as ratios of integers. (personal.math.ubc.ca)

Distance and generalizations

In analytic geometry, the theorem gives the Euclidean distance between points with perpendicular coordinate axes:

Repeated application extends this expression to three or more dimensions by summing the squared coordinate differences. (personal.math.ubc.ca)

For an arbitrary Euclidean triangle, the law of cosines states

When the included angle $C$ is $90^\circ$, its cosine vanishes, recovering the Pythagorean theorem. This connects the result with trigonometry. (personal.math.ubc.ca)

In real inner product spaces, including those studied in linear algebra, orthogonal vectors satisfy

The identity follows by expanding the inner product and using $\langle u,v\rangle=0$. For nonorthogonal vectors, the expansion contains the additional term $2\langle u,v\rangle$; simply adding squared lengths is therefore not generally valid. (personal.math.ubc.ca)