CIRCLES

Notes on Circles

9.1 Angle Subtended by a Chord at a Point

• Definition: The angle subtended by a line segment (chord) PQ at a point R not on the line is denoted as ∠PRQ. This concept extends to angles subtended at the center of the circle.
• Observations: Within a circle:
  • Longer chords subtend larger angles at the center.
  • Equal chords subtend equal angles at the center.

Theorem 9.1

• Statement: Equal chords subtend equal angles at the center.
• Proof: Using congruent triangles formed by radii and equal chords, it proves that the angles ∠AOB and ∠COD are equal.

Theorem 9.2

• Statement: If two chords subtend equal angles at the center, then the chords are equal.
• This theorem acts as a converse to Theorem 9.1.

9.2 Perpendicular from the Centre to a Chord

• Theorems:
  • Theorem 9.3: The perpendicular from the center of a circle to a chord bisects the chord.
    • This is proven through congruency of right triangles formed with radii and segments created by the bisector.
  • Theorem 9.4: The line drawn from the center to bisect a chord is perpendicular to that chord.

9.3 Equal Chords and their Distances from the Centre

• Distance Definition: The shortest distance from a point to a chord is the length of the perpendicular dropped from that point to the chord.
• Drawing equal chords in a circle reveals that longer chords are closer to the center than shorter chords, leading to:
  • Theorem 9.5: Equal chords are equidistant from the center of the circle.
  • Theorem 9.6: Chords that are equidistant from the center are equal in length.

9.4 Angle Subtended by an Arc

• Concept: The angle subtended at the center by an arc is equal to the angle subtended by the chord defining that arc.
• Theorem 9.7: The angle subtended by an arc at the center is double that subtended at any point on the circle.
• Theorem 9.8: Angles in the same segment of a circle are equal.

9.5 Cyclic Quadrilaterals

• Definition: A cyclic quadrilateral is one where all vertices lie on the circumference of a circle.
• Theorem 9.10: The sum of opposite angles of a cyclic quadrilateral equals 180º.
• Theorem 9.11: If the sum of a pair of opposite angles of a quadrilateral equals 180º, the quadrilateral is cyclic.

Important Examples

• Various practical examples demonstrating the theorems, such as proving angles equal or verifying properties of cyclic quadrilaterals using intersection and angle properties, reinforce learning.

Summary of Key Concepts

1. A circle consists of all points equidistant from a fixed point.
2. Equal chords subtend equal angles at the center.
3. Perpendiculars from the center bisect chords.
4. Chords equidistant from the center are equal in length.
5. Angles subtended by arcs at the center have specific relationships with angles at other points in the circle.
6. Cyclic quadrilaterals have important properties regarding angles.
7. The concepts can be applied in various geometrical and algebraic contexts to solve problems involving chords, angles, and cyclic figures.

1. A circle is made up of all points equidistant from a center. 2. Equal chords subtend equal angles at the center of the circle. 3. The perpendicular from the center to a chord bisects that chord. 4. Equal chords are equidistant from the center. 5. Angles in the same segment of a circle are equal. 6. The angle at the center is double the angle at any point on the circle subtended by the same arc. 7. The sum of opposite angles in a cyclic quadrilateral is 180°. 8. Chords equidistant from the center of the circle are equal in length.