Circles: Class 9 Study Guide
A circle is all points at the same distance from a centre. Its chords and angles follow several fixed rules.
A circle is all the points in a plane at a fixed distance, the radius, from a fixed point, the centre. A chord joins two points on the circle. The longest chord is the diameter. An arc is a part of the circle, and a segment and a sector are regions bounded by a chord or radii.
Several useful theorems link chords, arcs and angles.
Main results
| Result | In words |
|---|---|
| Equal chords | Subtend equal angles at the centre, and are equally far from it |
| Perpendicular from the centre | Bisects the chord |
| Angle at the centre | Is double the angle at the circumference on the same arc |
| Angles in the same segment | Are equal |
| Angle in a semicircle | Is a right angle |
| Cyclic quadrilateral | Opposite angles add up to 180° |
A chord problem
A chord is 8 cm long and is 3 cm from the centre. The perpendicular from the centre bisects the chord, so a right-angled triangle forms with sides 3 cm and 4 cm. The radius is its hypotenuse: √(9 + 16) = √25 = 5 cm.
It bisects the chord.
Half chord and distance.
Find the radius.
The radius must be longer than each leg.
Cyclic quadrilaterals
A quadrilateral whose four vertices lie on a circle is cyclic. Its opposite angles add up to 180°. If one angle is 75°, the opposite angle is 105°. If an angle at the centre is 120°, the angle at the circumference on the same arc is 60°.
Worked examples and common mistakes
Example: A diameter subtends an angle at the circumference. It is 90°. Example: The angle at the centre is 100°, so the angle at the circumference on the same arc is 50°. Common mistakes: halving when you should double, using the theorem for angles on different arcs, and treating a quadrilateral as cyclic without all four points on the circle.
Check your understanding: model answers
1. Half the chord is 12, so the radius is √(144 + 25) = 13 cm. 2. 70°. 3. 82°.
✨ Key Takeaways
- A perpendicular from the centre bisects a chord.
- The angle at the centre is double the angle at the circumference on the same arc.
- An angle in a semicircle is 90°.
- Opposite angles of a cyclic quadrilateral add up to 180°.
💬 Questions for Discussion & Reflection
- A chord of 24 cm is 5 cm from the centre. Find the radius.
- The angle at the centre is 140°. Find the angle at the circumference on the same arc.
- One angle of a cyclic quadrilateral is 98°. Find the opposite angle.