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In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
A square can be divided into an even number of triangles of equal area (left), but into an odd number of only approximately equal area triangles (right). Monsky's proof combines combinatorial and algebraic techniques and in outline is as follows: Take the square to be the unit square with vertices at (0, 0), (0, 1), (1, 0) and (1, 1).
The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.
In a right triangle there are two inscribed squares, one touching the right angle of the triangle and the other lying on the opposite side. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side. [38] An inscribed square can cover at most half the area of the triangle it is inscribed ...
Start with a square. (The black square in the image) Image 2: At each convex corner of the previous image, place another square, centered at that corner, with half the side length of the square from the previous image. Take the union of the previous image with the collection of smaller squares placed in this way. Images 3–6: Repeat step 2.
By the Wallace–Bolyai–Gerwien theorem, a square can be cut into parts and rearranged into a triangle of equal area. In geometry , the Wallace–Bolyai–Gerwien theorem , [ 1 ] named after William Wallace , Farkas Bolyai and P. Gerwien , is a theorem related to dissections of polygons .
The propositions in Book I concern the properties of triangles and parallelograms, including for example that parallelograms with the same base and in the same parallels are equal and that any triangle with the same base and in the same parallels has half the area of these parallelograms, and a construction for a parallelogram of the same area ...
Mathematically, this can be written as + =, where a is the length of one leg, b is the length of another leg, and c is the length of the hypotenuse. [ 2 ] For example, if one of the legs of a right angle has a length of 3 and the other has a length of 4, then their squares add up to 25 = 9 + 16 = 3 × 3 + 4 × 4.