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Subjects

Mathematics

Circles

In Class 9 Mathematics, the chapter Circles introduces the basic parts and properties of a circle, including its centre, radius, diameter, chord, arc and sector. Students explore cyclic points and learn how angles are subtended by chords. The chapter also explains the circumcircle of a triangle and the role of its circumcentre, building geometric understanding through definitions, diagrams and relationships within circles.

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Centre

In Class 9 Mathematics, this topic introduces the centre of a circle as the fixed point that is equidistant from every point on its circumference. Students learn how the centre connects to radii, diameters and chords, and how it helps identify key properties of circles. They use these ideas to understand why the perpendicular from the centre to a chord bisects the chord and how equal chords relate to equal distances from the centre, supporting clear geometric reasoning in the Circles chapter.

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Radius

In this Class 9 Mathematics topic from the chapter Circles, students learn that the radius is the line segment joining the centre of a circle to any point on its circumference. They explore why all radii of the same circle are equal, use the symbol r, and understand its connection with the diameter through d = 2r. The topic also supports solving problems involving a circle’s circumference and area, while strengthening visual reasoning and accurate use of geometric measurements.

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Diameter

In Class 9 Mathematics, the topic Diameter introduces the line segment joining two points on a circle and passing through its centre. Students learn that the diameter is the longest chord of a circle and that its length is twice the radius, while the radius is half the diameter. They use these relationships to find unknown lengths, interpret circle diagrams, and solve geometry problems involving centres, chords, and measurements.

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Chord

In Class 9 Mathematics, this topic introduces the chord of a circle—a line segment joining any two points on its circumference. Students learn to identify and draw chords, distinguish them from the radius, diameter, arc, and secant, and understand how a diameter is the longest chord. The topic also develops understanding of chords through diagrams and circle-based reasoning, helping students apply the concept while studying the properties of circles.

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Arc

In Class 9 Mathematics, this topic introduces an arc as a part of a circle’s circumference bounded by two points. Students learn to identify arcs in diagrams, distinguish between minor and major arcs, and relate an arc to its endpoints and the corresponding central angle. The lesson strengthens visual understanding of the Circles chapter and supports accurate interpretation of geometric figures and circle-based problems.

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Sector

In Class 9 Mathematics, the Circles chapter introduces the sector as the region enclosed by two radii and the arc between them. Students learn to identify minor and major sectors, relate a sector’s central angle to the complete angle at the centre, and connect it with the corresponding arc, circumference, and area of the circle. They use these relationships to interpret diagrams and solve problems involving sector angles, arc lengths, and areas.

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Cyclic points

In Class 9 Mathematics, Cyclic points introduces the idea of two or more points lying on the circumference of the same circle, called concyclic points. Students learn to identify such sets in diagrams, understand how a circle may pass through them, and distinguish points on the circle from points inside or outside it. The topic strengthens geometric reasoning within the Circles chapter and supports the interpretation of related figures, chords, and angle constructions.

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Angle subtended by chord

In Class 9 Mathematics, this topic from the chapter Circles explains how a chord forms an angle at a point on the circle and at the centre. Students learn to identify the angle subtended by a chord, understand the relationship between the central angle and the angle subtended by the same chord at the circumference, and apply the result that angles in the same segment are equal. They also use diagrams and simple reasoning to solve circle-based problems, including cases involving a semicircle and a right angle.

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Circumcircle

In this Class 9 Mathematics topic from the chapter Circles, students learn that a circumcircle is a circle passing through all the vertices of a triangle or other polygon. They explore the circumcenter, the point where the perpendicular bisectors of the sides meet, and understand how it determines the circle’s radius. The topic also connects equal distances from the circumcenter with chords, angles, and basic geometric constructions.

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Circumcentre

In Class 9 Mathematics, the topic Circumcentre introduces the point where the perpendicular bisectors of a triangle’s sides meet. Students learn that this point is equally distant from all three vertices and serves as the centre of the triangle’s circumcircle. Within the Circles chapter, they connect this idea with circle properties, geometric constructions, diagrams, and problems involving distances, radii, and the position of the circumcentre in different types of triangles.

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