Subtends

Subtends is a geometric term describing the relation between a line segment or chord and a point or set of points. A line segment AB subtends an angle at a point P if the rays PA and PB form the angle ∠APB. In circle geometry, a chord AB subtends an angle at any point on the circle’s circumference; this angle is known as the inscribed angle subtending AB. All inscribed angles subtending the same chord AB are equal when measured from points on the same arc.

Let O be the center of the circle and ∠AOB the central angle subtending AB. Then, for any inscribed angle ∠APB subtending AB, ∠APB = ∠AOB/2. Consequently, the central angle is twice the inscribed angle. If R denotes the circle’s radius, the chord length AB equals 2R sin(∠APB) = 2R sin(∠AOB/2). The arc AB subtends the central angle ∠AOB, and its length is s = R·∠AOB (with the angle in radians).

Beyond circles, subtend describes how a segment or curve defines an angle at a given point; equal chords subtend equal angles from the same circle, and the concept extends to broader Euclidean contexts where a segment or arc forms the sides of an angle at a vertex.