Angle Subtended by a Chord

Definition: The angle subtended by a chord at a point is the angle formed by the two line segments drawn from the endpoints of the chord to that point.

Central Angle Theorem:

Angle subtended by a chord at the centre = 2 × Angle subtended at any point on the remaining circle

If chord PQ subtends ∠POQ at centre O and ∠PRQ at point R on the circle, then:

∠POQ = 2 × ∠PRQ

Key terms:

• Central angle: The angle subtended by the chord at the centre of the circle.
• Inscribed angle: The angle subtended by the chord at any point on the circle.
• Arc: The part of the circle cut off by a chord. A chord creates a major arc and a minor arc.
• Segment: The region between a chord and the arc it cuts off.

Important:

• The point R must be on the same arc (on the opposite side of the chord from the centre for the minor arc case).
• If R is on the major arc, ∠PRQ is the angle in the major segment.
• If R is on the minor arc, the theorem still holds but the central angle is reflex.

Proof: Angle subtended by a chord at the centre is twice the angle at any point on the circle

Given: Circle with centre O. Chord PQ subtends ∠POQ at centre and ∠PRQ at point R on the major arc.

To prove: ∠POQ = 2 ∠PRQ

Construction: Join RO and extend it to a point S beyond O.

Case 1: Centre O lies inside ▵PRQ

1. In ▵POR: OP = OR (radii). So ▵POR is isosceles.
2. Let ∠OPR = ∠ORP = α
3. ∠POS = ∠OPR + ∠ORP = 2α (Exterior angle of ▵POR) …(i)

1. In ▵QOR: OQ = OR (radii). So ▵QOR is isosceles.
2. Let ∠OQR = ∠ORQ = β
3. ∠QOS = ∠OQR + ∠ORQ = 2β (Exterior angle of ▵QOR) …(ii)

Adding (i) and (ii):

• ∠POS + ∠QOS = 2α + 2β = 2(α + β)
• ∠POQ = 2(α + β) = 2 ∠PRQ ■

Case 2: Centre O lies outside ▵PRQ

• The same argument applies, but with subtraction instead of addition.
• ∠POQ = ∠QOS − ∠POS = 2β − 2α = 2(β − α) = 2 ∠PRQ ■

Case 3: R is such that RO passes through P or Q

• This reduces to one isosceles triangle, and the result follows directly from the exterior angle theorem.

Different Configurations of the Theorem:

1. Point R on the major arc (standard case)

• ∠PRQ is an acute angle (if PQ is not a diameter).
• ∠POQ = 2 ∠PRQ.
• This is the most common configuration in textbook problems.

2. Point R on the minor arc

• ∠PRQ is an obtuse angle.
• The central angle on the same side as R is the reflex angle at O.
• Reflex ∠POQ = 2 ∠PRQ.

3. PQ is a diameter

• ∠POQ = 180° (straight line through centre).
• ∠PRQ = 180/2 = 90° for any point R on the circle.
• This is Thales' Theorem (angle in a semicircle).

4. Equal chords

• Equal chords subtend equal central angles.
• Therefore, they subtend equal inscribed angles at any point on the corresponding arcs.

5. Chord through the centre (diameter)

• The longest chord of a circle is its diameter.
• Central angle = 180° (straight angle).

Applications of the Central Angle Theorem:

• Surveying and navigation: The inscribed angle property helps determine positions. An observer on the circumference of a circular path subtends a predictable angle to landmarks.
• Satellite communication: Satellites in geostationary orbit subtend angles at ground stations. The relationship between central and inscribed angles governs coverage areas.
• Optics and lens design: Light passing through circular lens elements follows angle relationships governed by the inscribed angle theorem.
• Proving other circle theorems: This theorem is the foundation for angles in the same segment, angle in a semicircle, and opposite angles of a cyclic quadrilateral.
• Architecture: Arched doorways and windows use circular arcs. The angle subtended at various points determines the visual perspective.
• Computer graphics: Circle algorithms in rendering use the inscribed angle relationship for smooth arc approximation.