source: ncert
topic: geometry
class_level: class_9
chapter: ch10
difficulty: intermediate
Class 9 | Ch10: Circles KEY DEFINITIONS: Circle: set of all points at equal distance (radius) from centre. Chord: line segment joining any two points on circle. Diameter: longest chord passing through centre = 2r. Arc: part of circle. Minor arc < semicircle. Major arc > semicircle. Sector: region between two radii and arc. Segment: region between chord and arc. THEOREMS: Theorem 1: Equal chords are equidistant from centre (and converse). Theorem 2: Equal chords subtend equal angles at centre (and converse). Proof: In triangles OAB and OCD (O=centre, AB=CD): OA=OC=OB=OD=radius. AB=CD (given). By SSS: triangles congruent. Angle AOB=Angle COD. Theorem 3: Perpendicular from centre bisects chord (and converse). Theorem 4: Angle in semicircle is 90°. Theorem 5: Angles in same segment are equal. Theorem 6: Angle at centre = double angle at circumference (same arc). Theorem 7: Opposite angles of cyclic quadrilateral sum to 180°. SOLVED EXAMPLES: Example 1: Chord AB=10cm. Distance from centre=12cm. Find radius. Perpendicular from centre bisects chord. Half chord=5cm. r^2=12^2+5^2=144+25=169. r=13cm. Example 2: Angle at centre=120°. Find angle at circumference (same arc). Angle at circumference=120/2=60°. Example 3: In cyclic quadrilateral, one angle=75°. Find opposite angle. Opposite angles sum=180°. Other angle=105°. COMMON MISTAKES: Angle at centre = TWICE angle at circumference (not half). Angle in semicircle=90° (angle subtended by diameter). Equal chords have equal angles at centre: equal angle does NOT mean equal chord unless same circle.