Class 9 th maths

Class 9 Mathematics: Chapter 5 - Circles
("I'm Up and Down, and Round and Round")

Overview: This chapter introduces the fundamental geometrical concepts, symmetries, theorems, and proofs related to circles, including chords, distances, arc subtensions, circumcircles, and cyclic quadrilaterals.

1. Definitions & Symmetries of a Circle

• Circle & Locus: A circle is the set (or locus) of all points in a two-dimensional plane that are equidistant from a fixed point called the centre. The fixed distance is the radius.
• Chord & Diameter: A line segment joining any two points on a circle is a chord. A chord passing through the centre is the diameter (the longest chord of a circle).
• Symmetry:
  • Rotational Symmetry: A circle has complete/infinite rotational symmetry about its centre (looks identical when rotated by any angle).
  • Reflection Symmetry: Every diameter acts as a line of reflection symmetry.

2. Circles Through Given Points

• Through 1 or 2 Points: Infinitely many circles can pass through one or two points. The centres of all circles passing through two points A and B lie on the perpendicular bisector of segment AB.
• Through 3 Points:
  • Collinear Points: No circle can pass through three collinear points.
  • Non-Collinear Points (Theorem 1): There is a unique circle passing through three non-collinear points.
• Circumcircle & Circumcentre: The unique circle passing through the three vertices of a triangle ΔABC is called its circumcircle, and its centre is the circumcentre (point of intersection of perpendicular bisectors of the sides).
  • Acute Triangle: Circumcentre lies inside the triangle.
  • Obtuse Triangle: Circumcentre lies outside the triangle.
  • Right Triangle: Circumcentre lies at the midpoint of the hypotenuse.

3. Chords and Angles Subtended at the Centre

Theorem 2: Equal chords of a circle subtend equal angles at the centre of the circle.

Theorem 3 (Converse): Chords of a circle that subtend equal angles at the centre are equal in length.

4. Midpoints and Perpendicular Bisectors of Chords

Theorem 4: The line segment joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.

Theorem 5 (Converse): The perpendicular dropped from the centre of a circle to a chord bisects the chord.

5. Distance of Chords from the Centre

Theorem 6: Chords of equal length in a circle are equidistant from the centre.

Theorem 7 (Converse): Chords that are equidistant from the centre of a circle are equal in length.

Theorem 8 (Unequal Chords): Given two unequal chords in a circle, the longer chord is closer to the centre (i.e., if AB > DE, then distance CF < CG).

Key Formula: For a chord of length L, radius r, and perpendicular distance d from centre:

Length of Chord = 2√(r² - d²)

6. Angles Subtended by Arcs

• Major Arc vs. Minor Arc: An arc subtending < 180° at the centre is a minor arc; an arc subtending > 180° is a major arc.

Theorem 9: The angle subtended by an arc at the centre of a circle is double the angle subtended by it at any point on the remaining part of the circle.

• Angles in the Same Segment: Angles subtended by an arc at any points in the same segment of a circle are equal (∠ADB = ∠AEB).
• Angle in a Semicircle: The angle subtended by a diameter at any point on the circle is a right angle (90°).

7. Concyclicity & Cyclic Quadrilaterals

• Concyclic Points: A set of points are called concyclic if they all lie on the boundary of the same circle.

Theorem 10: If a line segment joining two points subtends equal angles at two other points lying on the same side of the segment, then the four points are concyclic.

Theorem 11: The sum of either pair of opposite angles of a cyclic quadrilateral is 180° (supplementary).

Theorem 12 (Converse): If the sum of a pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic.

• Exterior Angle Property: The exterior angle at any vertex of a cyclic quadrilateral is equal to the interior opposite angle.

Quick Theorem Checklist

Concept	Property / Theorem
3 Non-collinear Points	Determines a unique circumcircle.
Equal Chords	Subtend equal central angles & are equidistant from centre.
Perpendicular from Centre	Bisects the chord.
Arc & Angle Theorem	Central angle = 2 × Angle at any point on remaining arc.
Angle in Semicircle	Always equal to 90°.
Cyclic Quadrilateral	Opposite angles sum to 180°.