CIRCLES

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circles and its properties

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Contents Circles a view What is a circle Parts of a circle Circle and centre of circle Circumference Circular region Radius Arc Diameter Concentric circle Semicircle Corollary theorems Cord Segments of circle Recap Crossword Exercises Test yourself - 1 Test yourself - 2

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CIRCLE IN DAILY LIFE A circle

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CIRCLE IN MUSIC Many musical instruments have a circular surface. For example: Bingo Drum Tabla Snare Drum Bass Drum

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CIRCLE IN SPORTS Five rings in the logo of Olympic games A circle

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A circle can be drawn with the help of a circular object. For example: A circle drawn with the help of a coin. A circle is a closed curve in a plane. What is a Circle?

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Parts of a Circle Segments & Lines chord diameter radius tangent secant

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This fixed point (equidistant) inside a circle is called centre. Circle and Centre of a circle A circle is a closed curve consisting of all points in a plane which are at the same distance (equidistant) from a fixed point inside it. O Centre A circle A circle has one and only one centre.

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The distance around a circle is called its circumference. O Centre Circumference A circle A

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Circular Region A circle divides a plane into three parts. 2. Interior of a circle 3. Exterior of a circle A plane O Centre The interior of a circle together with its circumference is called the circular region. 1. The circle

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Radius A line segment that joins any point on the circle to its centre is called a radius. M A point on the circle Radius Centre O (Contd…)

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Radii ( plural of radius ) of a circle are equal in length . Infinite number of radius can be drawn in a circle. Radius Centre K O L M N (Contd…)

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Diameter AB A line segment that joins any two points on the circle and passes through its centre is called a diameter . A B A circle O Centre Diameter (Contd…)

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A circle O M Infinite number of diameters can be drawn in a circle. As the radii of a circle are equal in length, its diameters too are equal in length. B Q (Contd…) Centre P A N

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The length of the diameter of a circle is twice the length of its radius. Radius OM Centre M O N Radius ON Diameter MN Diameter MN = Radius OM + Radius ON Radius OM = Radius ON (Contd…)

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Formulas Radius/diameter Circumference radius = ½diameter r = ½ d diameter = 2(radius) d = 2r C = 2 ∏r or C = ∏d

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Chord A line segment that joins any two points on the circle is called a chord . O B A A is a point on the circle B is another point on the circle A line segment that joins point A and B Chord

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Diameter is also a chord of the circle. O Chord CD C D M N K L Chord MN Chord KL (Contd…) Diameter CD

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The diameter is the longest chord. O Diameter CD C D M N Chord MN (Contd…) Chord KL L K

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L K M N Chord MN O Centre Infinite number of chords can be drawn in a circle. Chord KL Chord GH G H

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O Centre Arc An arc is the distance between any two points on the circumference of a circle. K L Arc (Contd…)

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O Centre L K An arc is named by three points , of which two are the end points of the arc and the third one lies in between them. X Naming an arc (Contd…) Arc KXL

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O Centre L K X Y An arc divides the circle into two parts: the smaller arc is called the minor arc , the larger one is called the major arc . Minor Arc KXL Major Arc KYL

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Example of an arc An arc An arc

No points of intersection (same center):

Concentric Circles No points of intersection (same center) Same center but different radii

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Half of a circle is called a semicircle. Centre O Diameter D E Semicircle S A semicircle is also an arc of the circle. R Arc DSE Semicircle DRE (Contd…) Semicircle DSE Arc DRE

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E Centre O Diameter Semicircle DSE Semicircle DRE Semicircular region Semicircular region The diameter of a circle divides it into 2 semicircular regions. D

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Segments of a Circle A chord divides the circular region into 2 parts , each of which is called a segment of the circle. Centre O D E Chord DE Minor segment of a circle Major segment of a circle S R (Contd…)

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Centre O D E Chord DE Minor segment of the circle Major segment of the circle P Q Minor arc DPE Major arc DQE The part of the circular region enclosed by a minor arc and the chord is called a minor segment. Minor segment does not contain the centre of the circle. The part of the circular region enclosed by a major arc and the chord is called a major segment. Major segment contains the centre of the circle.

An ANGLE on a chord:

An ANGLE on a chord An angle that ‘sits’ on a chord does not change as the APEX moves around the circumference … as long as it stays in the same segment We say “Angles subtended by a chord in the same segment are equal” Alternatively “Angles subtended by an arc in the same segment are equal” From now on, we will only consider the CHORD, not the ARC

Angle at the centre:

Angle at the centre Consider the two angles which stand on this same chord Chord What do you notice about the angle at the circumference? It is half the angle at the centre We say “ If two angles stand on the same chord, then the angle at the centre is twice the angle at the circumference ” A

Angle at the centre:

Angle at the centre We say “ If two angles stand on the same chord, then the angle at the centre is twice the angle at the circumference ” It’s still true when we move The apex, A, around the circumference A As long as it stays in the same segment 136° 272° Of course, the reflex angle at the centre is twice the angle at circumference too!!

Angle at Centre:

Angle at Centre A Special Case When the angle stands on the diameter, what is the size of angle a? a a The diameter is a straight line so the angle at the centre is 180° Angle a = 90° We say “The angle in a semi-circle is a Right Angle” A

A Cyclic Quadrilateral:

A Cyclic Quadrilateral …is a Quadrilateral whose vertices lie on the circumference of a circle Opposite angles in a Cyclic Quadrilateral Add up to 180° They are supplementary We say “Opposite angles in a cyclic quadrilateral add up to 180°”

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Radius Diameter Chord Arc Semi Circle Centre O Recap the Terms

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Radius Diameter Chord Arc Semi Circle Radius OM Centre M O

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Radius Diameter Chord Arc Semi Circle Centre E D Diameter DE O

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Radius Diameter Chord Arc Semi Circle Centre Chord PQ P Q O

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Radius Diameter Chord Arc Semi Circle E G Arc PQR O F Centre

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Radius Diameter Chord Arc Semi circle S Centre O Diameter Semicircle D E Semicircle DSE Semicircle

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C 2 C U M F E R N C E A E I Down 1. The distance between any two points on the circumference of the circle. 2. The distance around the circle. 3. The distance from the centre of the circle to a point on the circle. R D I U S R 1 C 3 R A Across: 4. The line segment that joins any two points on the circle and passes through its centre. 5. A closed curve in a plane. 6. All points on the circle are equidistant from this point. 7. A line segment that joins any two points on a circle. 4 D A M T E E 5 I R L E 6 C E N T E H R O D 7 Circle Crossword

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EXERCISE Question 1. Find the value of^AXB.(AOB=40degree) A B X O (i) A B O X (ii)

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Answer 1.i If ^AOB = 40 degree. Then ^AXB =1/2 x 40 = 20 . Therefore ^AXB =20 degree.

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ii If ^AOB = 40 degree Then ^AXB= 2 x 40 = 80. Therefore ^AXB = 80 degree HERE ^ IS USED FOR ANGLE SIGN.

ANSWER IN BRIEF:

ANSWER IN BRIEF How can the angle subtended by an arc at the centre be defined? The angle in semicircle is ______ degree. The angle subtended by an arc at the centre is ______ the angle subtended by it at any point on the circle. Which is the theorem that defines the relationship between the angles subtended by an arc at a centre and at a point on the circle ? P.S. For answers view the slideshow again

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The distance from the centre to the edge The distance from one side to the other passing through the centre The distance all of the way round the edge The blue line Where can you see i) a segment ii) a sector iii) an arc? Sector Segment An ARC is the name for part of the circumference RADIUS DIAMETER CIRCUMFERENCE CHORD

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12 32 9 6 D = ? r = ? r = ? D = ? 24 16 4.5 12

Use P to determine whether each statement is true or false.:

Use P to determine whether each statement is true or false . P Q R T S

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To APPROXIMATELY find the CIRCUMFERENCE MULTIPLY the DIAMETER by 3 (C = 3 x d) Radius Diameter Circumference 4 8 12 10 5 15 18 30 42

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Radius Diameter Circumference 2 4 12 4 8 24 6 12 36 10 20 60 5 10 30 15 30 90 3 6 18 5 10 30 7 14 42 To APPROXIMATELY find the CIRCUMFERENCE MULTIPLY the DIAMETER by 3 (C = 3 x d)

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Created by DEEPAK SHARMA KAPIL GYANPEETH The End

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