Class 9 Mathematics: Circles

• Introduction & Basic Concepts

  • Definition of a Circle
  • Radius, Diameter, and Chord
  • Secant and Tangent (basic distinction)
  • Arc (Minor and Major) and Semi-circle
  • Segment (Minor and Major)
  • Sector (Minor and Major)
  • Interior and Exterior of a circle
  • Concentric Circles

• Angle Subtended by a Chord at a Point

  • Theorem 1: Equal chords subtend equal angles at the center
  • Theorem 2: Converse of Theorem 1 (Chords subtending equal angles are equal)

• Perpendicular from the Centre to a Chord

  • Theorem 3: Perpendicular from the centre to a chord bisects the chord
  • Theorem 4: Line drawn through the centre to bisect a chord is perpendicular to the chord
  • Perpendicular Bisector of chords pass through the centre

• Circle Through Three Points

  • Collinear vs Non-collinear points
  • Theorem 5: One and only one circle passes through three non-collinear points
  • Circumcircle and Circumradius of a triangle

• Equal Chords and Their Distance from the Centre

  • Theorem 6: Equal chords are equidistant from the centre
  • Theorem 7: Chords equidistant from the centre are equal in length

• Angle Subtended by an Arc of a Circle

  • Theorem 8: Angle subtended by an arc at the centre is double the angle subtended at any point on the remaining part of the circle
  • Theorem 9: Angles in the same segment of a circle are equal
  • Theorem 10: Angle in a semi-circle is a right angle ($90^\circ$)

• Cyclic Quadrilaterals

  • Definition: All four vertices lie on the circle
  • Theorem 11: Sum of either pair of opposite angles of a cyclic quadrilateral is $180^\circ$ (supplementary)
  • Theorem 12: Converse of Theorem 11 (If opposite angles are supplementary, the quadrilateral is cyclic)
  • Exterior Angle Property: Exterior angle equals the interior opposite angle