✨ Geometry Mastery: Angles Subtended by Chords

Chord: A line segment connecting two points on a circle.

Angle Subtended: The angle formed at a point by two lines drawn from that point to the endpoints of a chord.

Theorem: The angle subtended by a chord at the center of a circle is twice the angle subtended at any point on the circumference.

Formula: ∠AOB = 2 × ∠ACB

Theorem: Equal chords of a circle subtend equal angles at the center.

Converse: If two chords subtend equal angles at the center, then the chords are equal in length.

Longer chords subtend larger angles at the center.

Shorter chords subtend smaller angles at the center.

Diameter subtends the largest possible angle (180°) at the center.

• Architecture: Designing circular structures

• Engineering: Calculating forces in circular components

• Navigation: Determining positions using circular geometry

Chord Length: c = 2r × sin(θ/2)

Central Angle: θ = 2 × arcsin(c/2r)

Circumference Angle: φ = θ/2