Circle Theorems II
4.8 Circle Theorems II – Cheat Sheet
1. Equal Chords are Equidistant from the Centre
- In a circle, chords of equal length are the same perpendicular distance from the centre.
- Converse: Chords equidistant from the centre are equal in length.
Example:
If AB and CD are chords and OA ⟂ AB, OC ⟂ CD, and OA = OC → AB = CD.
2. Perpendicular Bisector of a Chord Passes Through the Centre
- The perpendicular bisector of any chord always passes through the centre of the circle.
- Useful for locating the centre when it is not marked.
Example:
Draw perpendicular bisectors of two chords → intersection is the circle’s centre.
3. Tangents from an External Point are Equal in Length
- If two tangents are drawn to a circle from the same external point, their lengths are equal.
- The tangents make equal angles with the line joining the point to the centre.
Example:
From point P, tangents touch circle at A and B → PA = PB.
4. Applications in Problems
| Given | Property Used | Conclusion |
|---|---|---|
| Two chords AB and CD are same length | Equal chords | Distances from centre to chords are equal |
| Line is perpendicular bisector of chord PQ | Perpendicular bisector property | Passes through centre |
| Tangents PT and PS from external point P | Tangent property | PT = PS, ∠OPT = ∠OPS |
5. Quick Problem Examples
Example 1:
Chords AB and CD are equal and distance from centre O to AB is 5 cm.
→ Distance from O to CD = 5 cm.
Example 2:
From point P, tangents PA and PB touch a circle. PA = 8 cm.
→ PB = 8 cm.
Example 3:
Find the centre of a circle given three points on circumference:
- Draw two chords, construct perpendicular bisectors → intersection = centre.