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Chapter Overview
इस chapter में circles की definition और basic properties से
शुरुआत होती है। आगे circle की reflection और rotational symmetry,
two points से गुजरने वाले circles, three non-collinear points की
circumcircle, chords, perpendicular bisectors, chord की centre से
दूरी, arcs, subtended angles और concyclic points तथा cyclic
quadrilaterals को समझाया गया है।
Chapter में theorems, proofs, constructions, drawing और
reasoning-based activities भी दी गई हैं।
Topics Covered
Circle and its Definition
Radius and Diameter
Chord of a Circle
Locus of Points
Reflection Symmetry
Rotational Symmetry
Circles through Two Points
Three Non-collinear Points
Circumcircle
Circumcentre
Chords and Angles
Equal Chords
Perpendicular Bisector of Chords
Midpoint of a Chord
Distance of Chords from Centre
Longer Chord and Centre
Minor and Major Arcs
Angles Subtended by an Arc
Angle in a Semicircle
Concyclicity of Points
Cyclic Quadrilateral
Opposite Angles of a Cyclic Quadrilateral
Converse of Cyclic Quadrilateral Theorem
Theorems and Proofs
⭐ Key Results to Remember
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A circle is the set of points at a fixed distance
from a fixed centre.
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All diameters are lines of reflection symmetry.
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Infinitely many circles can pass through two given points;
their centres lie on the perpendicular bisector of the segment.
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Three non-collinear points determine a unique circle,
called the circumcircle.
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Equal chords subtend equal angles at the centre.
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The line joining the centre and midpoint of a chord is
perpendicular to the chord.
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Chords of equal length are equidistant from the centre.
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A longer chord is closer to the centre than a shorter chord.
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The angle subtended by an arc at the centre is twice the
angle subtended by the same arc at a point on the remaining circle.
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The angle subtended by a diameter at any point on the circle is 90°.
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Opposite angles of a cyclic quadrilateral add up to 180°.
Quick Revision
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Diameter is a chord passing through the centre.
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Every diameter is a line of reflection symmetry.
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Circumcentre is at the intersection of perpendicular
bisectors of the sides of a triangle.
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Equal chords ⇔ equal central angles.
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Centre to midpoint of a chord is perpendicular to the chord.
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Longer chord ⇔ closer to the centre.
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Central angle = 2 × angle at a point on the circle
subtended by the same arc.
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Angle in a semicircle = 90°.
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Cyclic quadrilateral: opposite angles sum to 180°.
📝 Exercise Structure
Official Chapter 5 में निम्न exercise sets और
End-of-Chapter Exercises दिए गए हैं:
Exercise Set 5.1
Exercise Set 5.2
Exercise Set 5.3
Exercise Set 5.4
Exercise Set 5.5
Exercise Set 5.6
End-of-Chapter Exercises
📚 Official NCERT Study Source
यह chapter page official NCERT Ganita Manjari Chapter 5
की definitions, theorems, activities और exercise structure
के आधार पर तैयार किया गया है।
Open Official Chapter 5 PDF →