Master the Pythagorean Theorem

Pythagorean Theorem and Its Proofs

• The Pythagorean theorem is a relation between the three sides of a right triangle that states the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

• The theorem is named after the Greek philosopher Pythagoras and has been proved numerous times using diverse methods, including geometric and algebraic proofs.

• In Euclidean space represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation.

• The theorem can be generalized to higher-dimensional spaces, spaces that are not Euclidean, objects that are not right triangles, and objects that are not triangles at all but n-dimensional solids.

• Proofs using constructed squares involve two squares with sides of length a+b and four right triangles whose sides are a, b, and c, arranged in a way to form a square in the center whose sides are length c.

• In another proof, rectangles in the second box can also be placed such that both have one corner that corresponds to consecutive corners of the square.

• Algebraic proofs involve four copies of the same triangle arranged symmetrically around a square with side c, which results in a larger square with side a+b and area (a+b)^2.

• Another algebraic proof uses four copies of a right triangle with sides a, b, and c arranged inside a square with side c, where the triangles are similar with area 1/2ab.

• Recent scholarship has cast increasing doubt on any role for Pythagoras as a creator of mathematics, although debate about this continues.

• The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power.

• Popular references in literature, plays, musicals, songs, stamps, and cartoons have been made to the Pythagorean theorem.

• The Pythagorean theorem is a fundamental concept in mathematics and is taught in schools worldwide.Pythagorean Theorem and its Proofs

• The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

• The Theorem may have more known proofs than any other, with 370 proofs compiled in the book 'The Pythagorean Proposition'.

• One proof is based on the proportionality of the sides of three similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.

• Einstein gave a proof by dissection in which the pieces do not need to be moved. Instead of using a square on the hypotenuse and two squares on the legs, one can use any other shape that includes the hypotenuse, and two similar shapes that each include one of two legs instead of the hypotenuse.

• Euclid's proof demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares.

• Another proof is given by rearrangement, where a large square is formed with area c2, from four identical right triangles with sides a, b, and c, fitted around a small central square.

• The converse of the theorem is also true, which states that if in a triangle, the square on one of the sides equals the sum of the squares on the remaining two sides, then the angle contained by the remaining two sides of the triangle is right.

• The theorem has important applications in trigonometry, geometry, calculus, and other fields of mathematics.

• The Pythagorean triples are three positive integers a, b, and c, such that a2 + b2 = c2, and they represent the lengths of the sides of a right triangle where all three sides have integer lengths.

• The inverse Pythagorean theorem relates the two legs of a right triangle with the length of the hypotenuse and the altitude.

• The theorem has been used in various fields including engineering, architecture, music, and astronomy.The Pythagorean Theorem and its Applications

• The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

• The Theorem can be used to find the distance between two points in Euclidean space by using the coordinates of the points.

• Incommensurable lengths can be constructed using the Theorem as the hypotenuse of a triangle is related to the sides by the square root operation.

• The Pythagorean School had difficulty dealing with incommensurable lengths because they only dealt with whole numbers and fractions.

• The Theorem can be used to find the distance of a complex number from zero in the complex plane.

• The Theorem can also be used to find the distance between two points in n-dimensional space.

• If Cartesian coordinates are not used, the formula for Euclidean distance becomes more complicated but can still be derived from the Theorem.

• The Pythagorean Trigonometric Identity relates the sine and cosine of an angle in a right triangle to the lengths of its sides.

• The cross product and dot product are related to the Theorem in a similar way.

• The Theorem has applications in optimization theory and statistics as it forms the basis of least squares.

• The Theorem is also used in physics to convert the straight-line distance between two points to curvilinear coordinates.

• The Pythagorean Theorem is a special case of the more general Law of Cosines, which is valid for arbitrary triangles.Generalizations of the Pythagorean Theorem

• The Pythagorean Theorem generalizes beyond the areas of squares on the three sides to any similar figures.

• The Law of Cosines is a more general theorem relating the lengths of sides in any triangle.

• Thābit ibn Qurra stated that the sides of three triangles were related as:

• Pappus's area theorem applies to triangles that are not right triangles using parallelograms on the three sides in place of squares.

• The Pythagorean Theorem can be applied to three dimensions to find the length of diagonal BD and diagonal AD.

• De Gua's theorem is a substantial generalization of the Pythagorean Theorem to three dimensions.

• The Pythagorean Theorem can be generalized to inner product spaces.

• The parallelogram law is a further generalization of the Pythagorean Theorem in an inner product space to non-orthogonal vectors.

• Another generalization of the Pythagorean Theorem applies to Lebesgue-measurable sets of objects in any number of dimensions.

• The Pythagorean Theorem is derived from the axioms of Euclidean geometry, and in fact, were the Pythagorean Theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean.

• Right triangles in a non-Euclidean geometry do not satisfy the Pythagorean Theorem.

• In hyperbolic geometry, the sides of the right triangle are shorter than predicted by the Pythagorean Theorem.