Concept

Hypotenuse

In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the other has a length of 4 (when squared, 16), then their squares add up to 25. The length of the hypotenuse is the square root of 25, that is, 5. The word hypotenuse is derived from Greek ἡ τὴν ὀρθὴν γωνίαν ὑποτείνουσα (sc. γραμμή or πλευρά), meaning "[side] subtending the right angle" (Apollodorus), ὑποτείνουσα hupoteinousa being the feminine present active participle of the verb ὑποτείνω hupo-teinō "to stretch below, to subtend", from τείνω teinō "to stretch, extend". The nominalised participle, ἡ ὑποτείνουσα, was used for the hypotenuse of a triangle in the 4th century BCE (attested in Plato, Timaeus 54d). The Greek term was loaned into Late Latin, as hypotēnūsa. The spelling in -e, as hypotenuse, is French in origin (Estienne de La Roche 1520). The length of the hypotenuse can be calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs (or catheti) of the triangle (the sides perpendicular to each other) are a and b and that of the hypotenuse is c, we have The Pythagorean theorem, and hence this length, can also be derived from the law of cosines by observing that the angle opposite the hypotenuse is 90° and noting that its cosine is 0: Many computer languages support the ISO C standard function hypot(x,y), which returns the value above. The function is designed not to fail where the straightforward calculation might overflow or underflow and can be slightly more accurate and sometimes significantly slower. Some scientific calculators provide a function to convert from rectangular coordinates to polar coordinates.

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Pythagorean theorem
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. The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation: The theorem is named for the Greek philosopher Pythagoras, born around 570 BC.
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
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A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle (ὀρθόσγωνία), is a triangle in which one angle is a right angle (that is, a 90-degree angle), i.e., in which two sides are perpendicular. The relation between the sides and other angles of the right triangle is the basis for trigonometry. The side opposite to the right angle is called the hypotenuse (side c in the figure).
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