kathets

Kathets is a rarely used plural form of kathetus, a term in Euclidean geometry referring to the two sides of a right triangle that meet at the right angle. In standard English geometry, these sides are more commonly called the catheti, with the singular cathetus. The word kathetus comes from the Greek kathetos meaning perpendicular; over time, math texts have preferred cathetus and catheti, though kathets may appear in some historical or translated works.

Definition and notation: In a right triangle, the two legs are the kathets or catheti. If the legs are labeled a and b, and the hypotenuse is c, then a^2 + b^2 = c^2 (the Pythagorean theorem). The lengths a and b determine the shape of the triangle and the values of corresponding trigonometric functions for the acute angles.

Relations to angles: For the angle A opposite leg a, sin A = a/c, cos A = b/c, and tan A = a/b. The analogous relations hold for the other acute angle with the legs swapped. The kathets are perpendicular to each other, forming the right angle of the triangle.

Applications: Kathets/catheti are fundamental in geometry and trigonometry. They are used to compute unknown side lengths, angles, or trigonometric ratios in right triangles, and appear in fields such as surveying, architecture, computer graphics, and various branches of mathematics.

See also: Right triangle, Hypotenuse, Cathetus, Catheti, Pythagorean theorem.