lhypoténuse

The lhypoténuse is the side of a right triangle opposite the right angle. It is the longest side and is typically denoted by c, while the other two sides are a and b. The lhypoténuse length is related to the legs by the Pythagorean theorem: a^2 + b^2 = c^2, so c = sqrt(a^2 + b^2). For example, in a 3-4-5 triangle, a = 3, b = 4, and c = 5.

Geometrically, the lhypoténuse has several notable properties: it is the diameter of the triangle’s circumcircle; by Thales’ theorem, any angle subtended by the diameter is a right angle. The lhypoténuse is the longest side, opposite the 90-degree angle.

In trigonometry, the lhypoténuse serves as the common reference for defining ratios: sin(theta) = opposite/lhypoténuse, cos(theta) = adjacent/lhypoténuse, for an acute angle theta. The tangent is expressed as opposite/adjacent, and tan(theta) = sin(theta)/cos(theta).

In applications, the lhypoténuse enables distance calculations in a plane: the distance between two points (x1, y1) and (x2, y2) equals the length of the lhypoténuse of the right triangle formed by the horizontal and vertical differences, i.e., sqrt((x2 − x1)^2 + (y2 − y1)^2).