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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.
Bride's Chair: Illustration of Pythagorean theorem Bride's Chair: The figure used by Euclid to explain the proof of Pythagorean theorem. In geometry, a Bride's Chair is an illustration of the Pythagorean theorem. [1] The figure appears in Proposition 47 of Book I of Euclid's Elements. [2]
Diagram to explain Garfield's proof of the Pythagorean theorem In the figure, A B C {\displaystyle ABC} is a right-angled triangle with right angle at C {\displaystyle C} . The side-lengths of the triangle are a , b , c {\displaystyle a,b,c} .
Experts described Jackson and Johnson’s approach as particularly challenging because trigonometry as a field is essentially based on Pythagoras’ theorem; thus using trigonometry to prove the ...
In this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem. The angle opposite the leg of length 1 (this angle can be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity ...
The celebrated Pythagorean theorem (book I, proposition 47) states that in any right triangle, 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 whose sides are the two legs (the two sides that meet at a right angle).
The use of the Pythagorean theorem and the tangent secant theorem can be replaced by a single application of the power of a point theorem. Case of acute angle γ, where a < 2b cos γ. Drop the perpendicular from A onto a = BC, creating a line segment of length b cos γ. Duplicate the right triangle to form the isosceles triangle ACP.
The Pythagorean theorem, and hence this length, can also be derived from the law of cosines in trigonometry. In a right triangle, the cosine of an angle is the ratio of the leg adjacent of the angle and the hypotenuse. For a right angle γ (gamma), where the adjacent leg equals 0, the cosine of γ also equals 0.