| --- |
| source: ncert |
| topic: geometry |
| class_level: class_9 |
| chapter: ch10 |
| difficulty: intermediate |
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| |
| 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. |
