Chapter 5: Circles
Explore chord properties, central angles, and the elegant rules of concyclicity.
š Core Definitions
Set of points equidistant from a fixed centre point.
A segment joining any two points on the boundary.
Half of a circle, divided equally by a diameter.
Multiple points lying on the exact same boundary.
A 4-gon whose vertices all lie on a single circle.
⚙️ Algorithm for Chord Length
Finding length using perpendicular distance.
1 Identify
Get circle's radius (r) & perpendicular distance (d).
2 Pythagoras
Half-Chord = √ r² − d² .
3 Double Length
Perpendicular bisects chord. Length = 2 × Half-Chord.
4 Finalise
Record final chord length with appropriate units.
š Essential Theorems
Angle in Semicircle = 90°
Centre Angle = 2 × Boundary
Cyclic Quad Opp Angles = 180°
⭐ Key Properties & Proofs
Perpendicular Bisector Rule:
The perpendicular drawn from the centre to a chord perfectly bisects the chord.
Equidistant Chords Rule:
Chords of equal length lie at strictly equal perpendicular distances from the centre.
The "Longer Chord" Rule:
Given two unequal chords, the longer chord is physically closer to the centre.
Angles in the Same Segment:
Angles subtended by an arc at different points on the remaining segment boundary are equal.
⚠️ Common Exam Traps
Trap 1: Linear Ratio Assumptions
Avoid assuming that if a chord is twice the distance from the centre, it is half the length. Always use Pythagoras.
Trap 2: The Half-Chord Oversight
After applying Pythagoras (√r²−d²), students often forget to double the value. Write "Chord = 2 × Half-Chord".