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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.
There are no Pythagorean triangles in which the hypotenuse and one leg are the legs of another Pythagorean triangle; this is one of the equivalent forms of Fermat's right triangle theorem. [12]: p. 14 Each primitive Pythagorean triangle has a ratio of area, K, to squared semiperimeter, s, that is unique to itself and is given by [22]
This proof is independent of the Pythagorean theorem, insofar as it is based only on the right-triangle definition of cosine and obtains squared side lengths algebraically. Other proofs typically invoke the Pythagorean theorem explicitly, and are more geometric, treating a cos γ as a label for the length of a certain line segment. [13]
Two New Orleans high school students have proven the Pythagorean Theorem using trigonometry without relying on circular reasoning. That should be impossible.
Then ABD is a 30°–60°–90° triangle with hypotenuse of length 2, and base BD of length 1. The fact that the remaining leg AD has length √ 3 follows immediately from the Pythagorean theorem. The 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression.
The triangle angle sum theorem states that the sum of the three angles of any triangle, in this case angles α, β, and γ, will always equal 180 degrees. The Pythagorean theorem states that the sum of the areas of the two squares on the legs ( a and b ) of a right triangle equals the area of the square on the hypotenuse ( c ).
Triangle with altitude h cutting base c into d + (c − d). By the Pythagorean theorem we have b 2 = h 2 + d 2 and a 2 = h 2 + (c − d) 2 according to the figure at the right. Subtracting these yields a 2 − b 2 = c 2 − 2cd. This equation allows us to express d in terms of the sides of the triangle:
The Greek word νυμφη has two relevant definitions: 'bride', and 'winged insect'. The figure of a right triangle with the three squares has reminded various writers of an insect, so the 'insect' sense of the Greek word came to be applied to right triangles with three squares, and to the Pythagorean theorem.