The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. There are many different proofs of this theorem, including algebraic proofs using similarity, geometric proofs comparing areas, and vector-based proofs. Pythagoras was a famous Greek mathematician and philosopher from Samos in the 6th century BC who is credited with discovering or proving the Pythagorean theorem, though some of his accomplishments may have been made by his colleagues or successors.

1. By Muhammad Hasan Bassari (20)

3. A. In mathematics, the Pythagorean theorem or Pythagoras' theorem is a statement about the
sides of a right triangle.
B. One of the angles of a right triangle is always equal to 90 degrees. This angle is the right
angle The two sides next to the right angle are called the legs and the other side is called the
hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the
longest side.
C. The Pythagorean theorem says that the area of a square on the hypotenuse is equal to the
sum of the areas of the squares on the legs. In this picture, the area of the blue square added
to the area of the red square makes the area of the purple square. It was named after the
greek MATHEMATICIANS PHYTAGORAS:
D. If the lengths of the legs are a and b, and the length of the hypotenuse is c, then, .
E. There are many different proofs of this theorem. They fall into four categories:
F. Those based on linear relations: the algebraic proofs.
G. Those based upon comparison of areas: the geometric proofs.
H. Those based upon the vector operation.

5.  An algebraic proof proceeds by similarity. It is a property of Right triangles, such as
the one shown in the above left figure, that the right triangle with sides , , and
(small triangle in the left figure; reproduced in the right figure) is similar to the
right triangle with sides , , and (large triangle in the left figure; reproduced in the
middle figure). Letting in the above left figure then gives

7. Pythagoras of Samos (Ancient Greek: Πυθαγόρας ὁ Σάμιος [Πυθαγόρης in
Ionian Greek] Pythagóras ho Sámios "Pythagoras the Samian", or simply
Πυθαγόρας; b. about 570 – d. about 495 BC)was an Ionian Greek
philosopher, mathematician, and founder of the religious movement called
Pythagoreanism. Most of the information about Pythagoras was written
down centuries after he lived, so very little reliable information is known
about him. He was born on the island of Samos, and might have travelled
widely in his youth, visiting Egypt and other places seeking knowledge.
Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and
there set up a religious sect. His followers pursued the religious rites and
practices developed by Pythagoras, and studied his philosophical theories.
The society took an active role in the politics of Croton, but this
eventually led to their downfall. The Pythagorean meeting-places were
burned, and Pythagoras was forced to flee the city. He is said to have
ended his days in Metapontum.
Pythagoras made influential contributions to philosophy and religious
teaching in the late 6th century BC. He is often revered as a great
mathematician, mystic and scientist, but he is best known for the
Pythagorean theorem which bears his name. However, because legend and
obfuscation cloud his work even more than that of the other pre-Socratic
philosophers, one can give only a tentative account of his teachings, and
some have questioned whether he contributed much to mathematics and
natural philosophy. Many of the accomplishments credited to Pythagoras
may actually have been accomplishments of his colleagues and successors.
Whether or not his disciples believed that everything was related to
mathematics and that numbers were the ultimate reality is unknown. It
was said that he was the first man to call himself a philosopher, or lover of
wisdom,and Pythagorean ideas exercised a marked influence on Plato, and
through him, all of Western philosophy.