Circles
Class 9 Maths • Chapter 09 • Comprehensive Guide
Why Study Circles?
- Used in wheels, clocks, coins, gears, fans
- Foundation for Areas of Circles (Class 10)
- Important geometry theorem-based chapter
- Very high weightage in proof + MCQs
1. Anatomy of a Circle
A circle is the collection of all points in a plane, which are at a fixed distance (radius) from a fixed
point (center).
Click parts of the circle:
Select a part...
Key Definitions (Write Exactly Like This)
- Circle: Set of all points in a plane at a fixed distance from a fixed point.
- Radius: Distance from centre to any point on the circle.
- Diameter: Longest chord of the circle (2 × radius).
- Chord: Line joining any two points on the circle.
- Arc: A part of the circumference.
2. Chords and the Center
Equal chords of a circle subtend equal angles at the center.
The perpendicular from the center of a circle to a chord bisects the chord.
Perpendicular Drop
Click "Drop" to see the theorem in action.
Circles – Theorem Map 🧠
- Equal chords subtend equal angles at the centre
- Equal chords are equidistant from the centre
- Perpendicular from centre bisects the chord
- Angle at centre = 2 × angle at circumference
- Angles in same segment are equal
- Angle in semicircle is 90°
- Opposite angles of cyclic quadrilateral are supplementary
3. Angles Subtended by an Arc
Theorem 9.8: The angle subtended by an arc at the center is double the angle subtended
by it at any point on the remaining part of the circle.
$$ \angle AOB = 2 \times \angle APB $$
Angle in the Same Segment
Theorem: Angles in the same segment of a circle are equal.
Exam Tip:
If two angles subtend the same chord from the same side,
they are equal.
Angle in a Semi-Circle
Theorem: The angle subtended by a diameter at any point on the circle is
90°.
If AB is diameter → ∠APB = 90°
4. Cyclic Quadrilaterals
A quadrilateral is called cyclic if all its four vertices lie on a circle.
Property: The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
Cyclic Quad Solver
If \( \angle A \) is given, find opposite \( \angle C \).
How to Write Proofs (CBSE Style)
- Write Given
- Write To Prove
- Join required lines (construction)
- Apply known theorem
- Conclude with statement
Always write theorem name before using it ✔
Common Mistakes to Avoid ❌
- ❌ Angle at centre is NOT equal to angle at circle
- ❌ All quadrilaterals are NOT cyclic
- ❌ Radius perpendicular to chord only at midpoint
- ❌ Diameter is the ONLY longest chord
Assertion–Reason Practice
Assertion (A): The angle in a semicircle is a right angle.
Reason (R): The angle subtended at the centre is twice the angle at the circle.
✔ A and R are true, and R explains A
Circles in Real Life 🌍
- Wheels rotate around a fixed centre
- Clock angles follow circle theorems
- Coins & plates are perfect circles
- Satellite paths use circular geometry
One-Page Revision
- Diameter = 2 × Radius
- Angle at centre = 2 × angle at circumference
- Angle in semicircle = 90°
- Same segment → equal angles
- Cyclic quad → opposite angles = 180°
Concept Mastery Quiz
1. The longest chord of a circle is called:
2. Angles in the same segment of a circle are:
3. The opposite angles of a cyclic quadrilateral sum to:
4. A perpendicular drawn from the center to a chord:
5. The angle in a semi-circle is: