Hypotenuse
Definition
The Hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The term comes from the Greek word 'hypoteinousa', meaning 'stretching under', which is fitting as the hypotenuse stretches across from one acute angle of the triangle to the other.
Mathematical Properties
The length of the hypotenuse can be found using the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This can be written as a^2 + b^2 = c^2, where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
Applications
The concept of the hypotenuse is fundamental in trigonometry, where the ratios of the lengths of the sides of a right-angled triangle are used to define the trigonometric functions sine, cosine, and tangent. These functions are used in a wide range of fields, including physics, engineering, computer science, and architecture.
The hypotenuse is also used in the calculation of distances. In the Cartesian coordinate system, the distance between two points can be found using the Pythagorean theorem, with the difference in x-coordinates and y-coordinates forming the two shorter sides of a right-angled triangle, and the distance between the points being the hypotenuse.
History
The concept of the hypotenuse and its properties were known to ancient civilizations, including the Egyptians and Babylonians. However, it was the ancient Greeks who provided the first known proof of the Pythagorean theorem. The proof is attributed to the Greek mathematician Pythagoras, although it is likely that the theorem was known to earlier mathematicians.