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📘 Class 9 Maths Chapter 9 – Circles | Full Chapter Summary

In this chapter, "Circles", we study the basic terms related to a circle, important theorems, and their proofs using triangle congruence and properties. This chapter forms the base for Class 10 Circle theorems and tangents.


🔹 1. Basic Terms Related to a Circle

Circle → Set of all points in a plane that are at a fixed distance (radius) from a fixed point (centre).

Radius (r) → Distance from centre to any point on the circle.

Chord → A line segment joining two points on the circle.

Diameter → Longest chord, passing through the centre.

Arc → A part of the circumference of a circle.

Segment → Part of circle bounded by a chord and its arc (Minor segment & Major segment).

Sector → Region enclosed by two radii and an arc (Minor sector & Major sector).

Tangent → A line that touches the circle at exactly one point.

Secant → A line that intersects the circle at two points.


🔹 2. Angle Properties of a Circle

Angle Subtended by a Chord at a Point → The angle made by chord AB at a point on the circle (say ∠ACB).

Perpendicular from Centre to Chord → Bisects the chord.

Equal Chords → Equal Angles at the Centre (and converse is true).


🔹 3. Important Theorems of Circles

Theorem 9.1
Equal chords of a circle subtend equal angles at the centre.

Theorem 9.2
If the angles subtended by two chords at the centre are equal, then the chords are equal.

Theorem 9.3
The perpendicular from the centre of a circle to a chord bisects the chord.

Theorem 9.4
The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

Theorem 9.5
Angles subtended by an arc of a circle at different points on the remaining part of the circle are equal.

Theorem 9.6
Angle subtended by a diameter at the circumference is a right angle.

Theorem 9.7
The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.


🔹 4. Cyclic Quadrilateral

A quadrilateral whose all four vertices lie on a circle.

Property → Opposite angles are supplementary.


🎯 Key Learning Outcomes

Understanding circle-related terms: chord, arc, sector, segment, tangent.

Applying theorems to find unknown angles and lengths in circles.

Proving properties using congruence of triangles.

Learning the concept of cyclic quadrilaterals and their angle property.

Strengthening the base for Class 10 circle theorems (tangents & secants).


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