kathetos

Kathetos is a term used in geometry to denote one of the two legs of a right triangle. The two legs, also called catheti, are the sides that meet at the right angle, while the third side is the hypotenuse. The lengths of the two catheti are typically denoted a and b, and the hypotenuse length is c, related by the Pythagorean theorem a^2 + b^2 = c^2.

Etymology and usage: the word kathetos comes from Greek (κάθετος), meaning perpendicular. The Latinized form cathetus appears in older mathematical texts, and the plural catheti is used when referring to both legs. In modern English, the term leg is more common, but cathetus (and its plural) remains standard in some mathematical literature.

Properties: the catheti are perpendicular to each other, forming the right angle. They determine the shape and size of the triangle in conjunction with the hypotenuse. In trigonometric terms, if an acute angle θ lies between one cathetus and the hypotenuse, then cos θ = adjacent/hypotenuse (the length of that cathetus divided by c), sin θ = opposite/hypotenuse (the other cathetus divided by c), and tan θ = opposite/adjacent = (the other cathetus)/(that cathetus).

Examples: in a 3-4-5 triangle, the catheti have lengths 3 and 4, while the hypotenuse is 5. Such triangles are common in teaching, design, and computer graphics due to their simple integer relationship.

See also: right triangle, hypotenuse, Pythagorean theorem, leg.