Circle Theorems I
4.7 Circle Theorems I – Cheat Sheet
1. Key Circle Theorems and Properties
| Theorem | Statement | Diagram Idea |
|---|---|---|
| Angle in a semicircle = 90° | Any triangle drawn with diameter as one side is right-angled at the circumference. | Diameter as base, angle opposite is 90°. |
| Tangent–radius = 90° | A tangent to a circle is perpendicular to the radius at the point of contact. | Radius meets tangent at 90°. |
| Angle at centre = 2 × angle at circumference | Angle formed at the circle’s centre is twice any angle formed at the circumference on the same arc. | Central angle double inscribed angle. |
| Angles in the same segment are equal | Angles subtended by the same arc at the circumference are equal. | Same arc → equal angles. |
| Cyclic quadrilateral opposite angles sum to 180° | In a quadrilateral with all vertices on the circle, opposite interior angles are supplementary. | A + C = 180°, B + D = 180°. |
| Alternate segment theorem | The angle between tangent and chord through the point of contact equals the angle in the opposite segment. | Tangent–chord angle = opposite inscribed angle. |
2. Using Circle Theorems to Find Angles
Example 1: Angle in semicircle
- Diameter AB, point C on circumference → ∠ACB = 90°.
Example 2: Tangent–radius
- Radius OA, tangent at A → ∠OAT = 90°.
Example 3: Angle at centre vs circumference
- Arc BC → ∠BOC = 2 × ∠BAC.
Example 4: Angles in same segment
- Arc DE → ∠DFE = ∠DGE.
Example 5: Cyclic quadrilateral
- ∠PQR = 110° → ∠PSR = 70°.
Example 6: Alternate segment
- Tangent at A, chord AB → ∠BAT = ∠ACB.
3. Quick Reference Table
| Given | Theorem | Conclusion |
|---|---|---|
| Diameter as one side of triangle | Angle in semicircle | Right angle opposite diameter |
| Tangent meets radius | Tangent–radius | 90° |
| Central and circumferential angles on same arc | Angle at centre twice circumference | Centre angle = 2 × circumference angle |
| Two angles subtended by same arc | Angles in same segment | Angles equal |
| Quadrilateral in circle | Cyclic quadrilateral | Opposite angles sum to 180° |
| Tangent and chord angle | Alternate segment theorem | Equals angle in opposite segment |