Circles - Angle Subtended by a Chord at a Point

Chord and Subtended Angle: A chord is a line segment joining any two points on the circumference of a circle. When the endpoints of a chord $AB$ are joined to a point $P$, the angle $\angle APB$ is called the angle subtended by the chord at point $P$. Visually, if $P$ is the center $O$, the chord forms the base of an isosceles triangle $\triangle AOB$.

Equal Chords and Center Angles: Equal chords of a circle (or of congruent circles) subtend equal angles at the center. If two chords $AB$ and $CD$ are equal in length ($AB = CD$), then the angles they form at the center $O$ are equal, meaning $\angle AOB = \angle COD$.

Converse of Equal Chords Theorem: If the angles subtended by two chords of a circle at the center are equal, then the chords themselves must be equal in length. This means if $\angle AOB = \angle COD$, then $AB = CD$.

Perpendicular from Center: The perpendicular line drawn from the center of a circle to a chord bisects the chord. Visually, if you draw a line from center $O$ meeting chord $AB$ at $M$ such that $\angle OMA = 90^{\circ}$, then $M$ is the midpoint of $AB$, so $AM = MB$.

Equidistant Chords: Equal chords of a circle are equidistant from the center. The 'distance' of a chord from the center is the length of the perpendicular segment from the center to the chord. Visually, if $AB = CD$, then the perpendicular distances $OX$ and $OY$ from the center $O$ to these chords will be equal ($OX = OY$).

Chord Length and Distance Relationship: As the length of a chord increases, its distance from the center decreases. The longest chord of a circle is the diameter, and its distance from the center is $0$.

Angle at the Circumference: For a fixed chord, the angle it subtends at the center is double the angle it subtends at any point on the remaining part of the circle. Visually, if a chord $AB$ subtends $\angle AOB$ at center $O$ and $\angle ACB$ at a point $C$ on the major arc, then $\angle AOB = 2\angle ACB$.