9.1 Angle Subtended by a Chord at a Point
- Chord: A line segment joining two points on a circle.
- Angle subtended by a chord: The angle formed at a point on the circle or outside the circle by the ends of a chord.
Key Points:
- Angle subtended at the centre is twice the angle at any point on the circle:
∠subtended at centre =2×∠subtended at circumference
- Angles in the same segment of a circle are equal.
9.2 Perpendicular from the Centre to a Chord
- Theorem: The perpendicular from the centre of a circle to a chord:
- Bisects the chord
- Is the shortest distance from centre to chord
Implication:
- If a chord is bisected, the line joining the centre and midpoint of chord is perpendicular to the chord.
9.3 Equal Chords and Their Distances from the Centre
- Theorem 1: Equal chords of a circle are equidistant from the centre.
- Theorem 2: Chords equidistant from the centre are equal in length.
Key Idea: Distance from centre to chord = perpendicular drawn from centre to chord.
9.4 Angle Subtended by an Arc of a Circle
- Arc: A part of the circumference of a circle.
- Angle subtended by an arc = angle formed at a point on the circumference.
- Properties:
- Angles in the same segment are equal
- Angle at centre = 2 × angle at circumference
9.5 Cyclic Quadrilaterals
- Definition: A quadrilateral is cyclic if all its vertices lie on a single circle.
- Property: Sum of opposite angles = 180°
∠A+∠C=180°,∠B+∠D=180°
Example: Rectangle and square are cyclic quadrilaterals.
Quick Short Q&A (Most Possible)
| Question | Short Answer |
|---|---|
| What is a chord? | Line joining two points on a circle |
| Angle subtended at centre? | Twice the angle at circumference |
| Angles in same segment? | Equal |
| Perpendicular from centre to chord? | Bisects the chord |
| Equal chords? | Equidistant from centre |
| Chords equidistant from centre? | Equal in length |
| Definition of arc? | Part of circumference |
| Angle subtended by an arc? | Angle formed at a point on circumference |
| Cyclic quadrilateral? | Quadrilateral with vertices on a circle |
| Sum of opposite angles in cyclic quad? | 180° |