# Circles

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Category: Education

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Circle

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Def:- A circle is defined as the collection of all the points on a plane that are at equal distance from a given fixed point on the plane. This fixed point is called as the centre of the circle and the fixed distance is called as the radius . Radius ‘r’ Centre ‘O’

### PowerPoint Presentation:

Radius : A line segment joining the centre of a circle with any point on its circumference is called the radius of the circle Chord : A line that joins two points on the circumference of a circle is called a chord . Diameter : A chord that passes through the centre of a circle is called the diameter of the circle . Diameter is the longest chord of a circle. The diameter of a circle is twice the radius . Semicircle : A diameter divides a circle into two equal parts, each is called a semicircle . Arc : The part of the circumference of a circle between two given points is called an arc

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Segments: A chord divides a circular area into two parts called segments they are major segment and minor segments . Sector: The region between two radii of a circle and any of the arcs between them is called a sector . The diameter of a circle divides it into two equal segments

### Chords of a Circle :

Chords of a Circle We know that the perpendicular from a point to a line segment is the shortest distance between them. A line that joins two points on the circumference of a circle is called a chord. A chord passing through the center of the circle is called diameter. The longest chord of the circle is the diameter .

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Theorem: The perpendicular from the centre of a circle to a chord bisects the chord. Theorem: The line drawn from the centre of a circle to bisect a chord is perpendicular to the chord

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Theorem: Equal chords of a circle are equidistant from the centre of the circle. Theorem: Chords equidistant from the centre of a circle are equal in length .

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Let AB be any chord of the circle with centre O. Then  AOB is called the angle subtended by the chord at the centre of the circle. As the chord moves away from the centre, its length and angle subtended by it at the centre decreases and, If it moves closer to the centre its length and angle subtended by it at the centre increases. Centre ‘O’

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Theorem: Chords that subtend equal angles at the centre of a circle are equal in length. Theorem: Equal chords of a circle subtend equal angles at the centre.

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Arc of a Circle A part of a circle is called an arc. Arcs of a circle that superimpose each other completely are called congruent arcs. If two arcs of a circle are congruent, then their corresponding chords are equal and conversely, if two chords of a circle are equal, then their corresponding arcs are congruent. CORRESPONDING ARCS OF TWO EQUAL CHORDS OF A CIRCLE ARE CONGRUENT

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Theorem: 1) Congruent arcs of a circle subtend equal angles at the centre and vice – versa 2) Equal chords cut off equal arcs and vice – versa TWO CONGRUENT ARCS AB AND CD THEN  AOB=  COD

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Theorem: The angle subtended by an arc at the centre is double the angle subtended by the arc at any point on the remaining circle.

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Angle in a Semi-Circle is a right angle

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Theorem: Angles subtended by an arc at all points within the same segment of the circle are equal.

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THANK YOU ! MADE BY, ADITYA SREEKUMAR CLASS – 9 TH ‘A’ ROLL NO. - 18 