Introduction Circle Some terms related with a circle


 Laurel Fisher
 4 years ago
 Views:
Transcription
1 141 ircle Introduction In our daytoday life, we come across many objects which are round in shape, such as dials of many clocks, wheels of a vehicle, bangles, key rings, coins of denomination ` 1, ` 2, ` 5, ` 10, etc. In a clock, we observe that second's hand goes round the dial of the clock rapidly and its tip moves in a round path. This path traced by the tip of the second's hand is called a circle. In this chapter, we shall study about circles, some terms related with a circle, chord properties of a circle, arc and chord properties of a circle ircle circle is the collection of all those points, say, in a plane each of which is at a constant distance from a fixed point in that plane. In other words, a circle is the path of a point which moves in a plane so that it remains at a constant distance from a fixed point in the plane. The fixed point is called the centre and the constant distance is called the radius of the circle. The radius of a circle is always positive. The adjoining figure shows a circle with as its centre and r as its ircle radius. ote that the centre of a circle does not lie on the circle. Let be the centre of a circle and r its radius. If is a point on the circle, then the line segment is a radius of the circle and its length is r. If is another point on the circle then is another radius of the circle. ote that all radii (plural of radius) have one point in common, which is the centre of the circle. lso = = r. Thus: ll radii of a circle are equal. ote. The line segment joining the centre and any point on the circle is also called a radius of the circle. Thus, radius is used in two senses in the sense of a line segment and also in the sense of its length Some terms related with a circle 1. ircleinterior and exterior circle is a closed curve. It divides the plane region into three parts. They are: (i) ircle The collection of all points of the plane such that = r form a circle with centre and radius r (r > 0). r entre r Radius
2 (ii) Interior of a circle The collection of all points of the plane which lie inside the circle i.e. the collection of all points of the plane such that < r form the interior of the circle. The collection of all points of the plane which either lie on the circle or are inside the circle form the circular region. (iii) Exterior of a circle The collection of all points of the plane which lie outside the circle i.e. the collection of all points of the plane such that > r form the exterior of the circle. Interior r r 2. hord of a circle a line segment joining any two points of a circle is called a chord of the circle. In the adjoining figure, is a chord of the circle with centre. The distance is called the length of the chord. 3. iameter of a circle chord of a circle which passes through its centre is called a diameter of the circle. diameter of a circle is the longest chord of the circle and all diameters have equal length. In the adjoining figure, is a diameter of the circle with centre. ote that and are both radii of the circle, so = = r. It follows that = 2r = 2 radius. Thus: Length of a diameter = 2 radius. ote that there are infinitely many lines passing through the point (centre of circle), so a circle has infinitely many diameters. Exterior hord iameter 4. Secant of a circle line which meets a circle in two points is called a secant of the circle. In the adjoining figure, line is a secant of the circle with centre. Secant 5. rc of a circle rc (continuous) part of a circle is called an arc of the circle. The arc of a circle is denoted by the symbol. In the adjoining figure, denotes the arc of the circle with centre. rc of a circle is divided into following categories: (i) inor and major arc n arc less than onehalf of the whole arc of a circle is called a minor arc of the circle and an arc greater than onehalf of the whole arc is called a major arc of the circle. Here, is the minor arc and R is the major arc. inor arc Understanding ISE mathematics Ix R ajor arc
3 Semicircle (ii) Semicircle When is a diameter, then both arcs are equal and each is called a semicircle. Thus, onehalf of the whole arc of a circle is called a semicircle. ircumference (iii) ircumference The whole arc of a circle is called the circumference of the circle. The length of the circumference of a circle is the length of the whole arc. However, in general, the term circumference of a circle refers to its length. 6. Sector of a circle The part of the plane region enclosed by an arc of a circle and its two bounding radii is called a sector of the circle. The part containing the minor arc is called minor sector and the part containing the major arc is called major sector. 7. Segment of a circle chord of a circle divides its circular region into two parts. Each part of the circular region is called a segment of the circle. The part of the circular region containing the minor arc is called a minor segment and the part containing the major arc is called a major segment. ajor sector inor sector inor segment ajor segment 8. ngle subtended by an arc The angle subtended by the two bounding radii of an arc of a circle at the centre of the circle is called the angle subtended by the arc. In the adjoining figure, is the angle subtended by the minor arc of a circle with centre and the reflex is the angle subtended by the major arc at. If R is any point on the major arc, then R is called the angle subtended by the minor arc at the point R. R 9. ngle subtended by a chord Let be a chord of a circle with centre and R, S be points on the minor and major arcs of the circle (as shown in the adjoining figure). Join and, then is the angle subtended by the chord at the centre. Join R, R, S and S, then R and S are the angles subtended by the chord at the points R and S respectively on the minor and major arcs. R S 10. oncentric circles Two or more circles are called concentric circles if and only if they have same centre but different radii. circle 1279
4 11. Equal (or congruent) circles Two or more circles are called equal (or congruent) circles if and only if they have same radius. In the adjoining figures, two circles with centres and have equal radius (= r), so, these are equal circles. r (a) r (b) 14.2 hord roperties of ircles Theorem 14.1 The straight line drawn from the centre of a circle to bisect a chord, which is not a diameter, is perpendicular to the chord. Given. chord of a circle with centre, and bisects the chord. To prove.. onstruction. Join and. roof. In s and 1. = 1. Radii of same circle. 2. = 2. is midpoint of. 3. = 3. ommon S.S.S. axiom of congruency. 5. Α = 5. c.p.c.t.. 6. Α + = is a st. line. 7. Α = 90 Hence..E.. 7. From 5 and 6. Theorem 14.2 (onverse of theorem 14.1) The perpendicular to a chord from the centre of the circle bisects the chord. Given. chord of a circle with centre, and is perpendicular to the chord. To prove. =. onstruction. Join and. roof. In s and 1. = 1. Radii of same circle. 2. = 2. Each = 90, since. 3. = 3. ommon R.H.S. axiom of congruency. 5. =.E.. 5. c.p.c.t Understanding ISE mathematics Ix
5 Theorem 14.3 Equal chords of a circle are equidistant from the centre. Given. and are chords of a circle with centre, and =. To prove. and are equidistant from i.e. if and, then =. onstruction. Join and. roof. 1. = erpendicular from centre bisects the chord (Theorem 14.2). 2. = Same as above. 3. = In s and 3. = (given). 4. = 4. From = 5. Each = 90, and. 6. = 6. Radii of same circle R.H.S. axiom of congruency. 8. =.E.. 8. c.p.c.t.. Theorem 14.4 (onverse of theorem 14.3) hords of a circle that are equidistant from the centre of the circle are equal. Given. and are chords of a circle with centre ;, and =. To prove. =. onstruction. Join and. roof. In s and 1. = 1. Given. 2. = 2. Radii of same circle. 3. = 3. Each = 90, and R.H.S. axiom of congruency. 5. = 5. c.p.c.t.. 6. = erpendicular from centre bisects the chord. 7. = Same as above = 1 2 =..E.. 8. =, from 5. circle 1281
6 Theorem 14.5 There is one and only one circle passing through three given noncollinear points. Given. Three noncollinear points, and. To prove. ne and only one circle can be drawn passing through the points, and. onstruction. Join and. raw perpendicular bisectors of line segments and, say and RS respectively. Let these bisectors meet at. Join, and. roof. R S 1. = 1. lies on the perpendicular bisector of and every point on the perpendicular bisector of a line segment is equidistant from its end points. 2. = 2. lies on the perpendicular bisector RS of and every point on the perpendicular bisector of a line segment is equidistant from its end points. 3. = = 3. From 1 and If a circle is drawn with as centre and radius =, it passes through the points and. 5. There is only one circle passing through the points, and. 4. From 3, is equidistant from points, and. 5. since two lines can intersect at only one point, the perpendicular bisectors of and meet at only one point. Hence, one and only one circle can be drawn passing through three given noncollinear points. orollary 1. In the plane of a circle, the perpendicular bisector of a chord of a circle passes through its centre. orollary 2. The perpendicular bisectors of two (nonparallel) chords of a circle intersect at the centre of the circle. orollary 3. s there is one and only one circle passing through three noncollinear points, two different circles can meet atmost in two different points. Illustrative Examples Example 1. chord of length 16 cm is drawn in a circle of diameter 20 cm. alculate its distance from the centre of the circle. Solution. Let be a chord of a circle with centre such that = 16 cm and radius of the circle = = 1. diameter = cm = 10 cm. 2 2 From, draw. Since, is bisected at (Theorem 14.2) = 1 = cm = 8 cm Understanding ISE mathematics Ix
7 From right angled, by ythagoras theorem, we get 2 = = = = = 36 = 6 cm. Hence, the given chord is at a distance of 6 cm from the centre of the circle. Example 2. The centre of a circle of radius 13 units is the point (3, 6). (7, 9) is a point inside the circle. is a chord of the circle such that =. alculate the length of. Solution. Given the centre of the circle is (3, 6) and (7, 9) is a point on the chord of the circle such that = i.e. is midpoint of the chord. Join. Since bisects the chord, (Theorem 14.1). Radius of the circle = = 13 units (given). (3, 6) = ( 7 3) 2 + ( 9 6) 2 istance formula = = = 5 units From right angled, by ythagoras theorem, we get 2 = = = = = 144 = 12 units. The length of chord = 2 = (2 12) units = 24 units. Example 3. In the figure given below, the diameter of a circle with centre is perpendicular to the chord. If = 8 cm and = 2 cm, find the radius of the circle. Solution. Since i.e., is bisected at (Theorem 14.2), = 1 = 1. 8 cm = 4 cm. 2 2 Let radius of circle = r cm, then = = (r 2). From right angled, by ythagoras theorem, we get 2 = r 2 = (r 2) 2 r 2 = 16 + r 2 4r + 4 4r = 20 r = 5. Radius of the circle = 5 cm. Example 4. Two chords, of lengths 24 cm, 10 cm respectively of a circle are parallel. If the chords lie on the same side of centre and the distance between them is 7 cm, find the length of a diameter of the circle. Solution. Let be the centre of the circle. raw and. = 1 = cm = 12 cm and = 1 = cm = 5 cm. Since, points, and are collinear. = distance between and = 7 cm (given). (7, 9) circle 1283
8 Let = x cm, then = + = (x + 7) cm. Let the radius of the circle be r cm. From right angled s and (by ythagoras theorem), we get 2 = and 2 = r 2 = (12) 2 + x 2 (i) and r 2 = (x + 7) 2 (ii) From (i) and (ii), we get (x + 7) 2 = (12) 2 + x x x + 49 = x 2 14x = 70 x = 5. From (i), r 2 = (12) = = 169 r = 13. The length of a diameter of the circle = 2r = 26 cm. Example 5. is an isosceles triangle inscribed in a circle. If = = 25 cm and = 14 cm, find the radius of the circle. Solution. Let be the centre of the circle. From, draw. Since is isosceles with =, is the midpoint of i.e. is perpendicular bisector of, therefore, lies on. = 1 = cm = 7 cm. 2 2 From right angled, by ythagoras theorem, we get 2 = 2 2 = (25) = = 576 = 24 cm. Let r cm be the radius of the circle, then = = (24 r) cm. From right angled, by ythagoras theorem, we get 2 = r 2 = (24 r) 2 r 2 = r 2 48r 48r = 625 r = = Hence, the radius of the circle = cm. Example 6. Two chords and of a circle are equal. rove that the centre of the circle lies on the bisector of. Given. and are chords of a circle with centre, and =. 1 To prove. lies on the bisector of i.e. =. onstruction. Join and. roof. In s and 284 Understanding ISE mathematics Ix 1. = 1. Radii of same circle. 2. = 2. Given.
9 3. = 3. ommon S.S.S. axiom of congruency. 5. =.E.. 5. c.p.c.t.. Example 7. In a circle of radius 5 cm, and are two chords such that = = 6 cm. Find the length of the chord. Solution. Let be the centre of the circle. Let the bisector of meet the chord at. In s and 1. = (by const.) 2. = (given) 3. = (common) = and Β = (S..S. axiom of congruency) ( c.p.c.t. ) ut + = 180 ( is a st. line) = 90. Therefore, is perpendicular bisector of, and hence it passes through the centre of the circle. Let = y cm and = x cm, then = = (5 x) cm [ radius = 5 cm] From right angled, y 2 + x 2 = 5 2 (i) From right angled, y 2 + (5 x) 2 = 6 2 i.e. y 2 + x 2 10 x = 11 (ii) Subtracting (ii) from (i), we get 10 x = x = 14 x = 7 5. Substituting this value of x in (i), we get y = 25 y2 = = 25 Length of chord = 2 = cm = 48 5 y 2 = cm = 9 6 cm. y = 24 5 Example 8. rove that the line joining midpoints of two parallel chords of a circle passes through the centre of the circle. Given., are two chords of a circle with centre., and are midpoints of and respectively. To prove. passes through. onstruction. Join, and through draw a straight line parallel to. roof m is midpoint of, and the line drawn from the centre of circle to bisect a chord is perpendicular to it. 2. = F E circle 1285
10 3. E = E, alternate angles are equal. 4. E = similarly, is midpoint of and same reasons as above. 5. E + E = dding 3 and is a straight line. Hence, passes through..e.. 6. Sum of adjacent angles is 180. Example 9. If two equal chords of a circle intersect, prove that their segments will be equal. Given. and are chords of a circle with centre. and intersect at and =. To prove. (i) = (ii) =. onstruction. raw,. Join. roof. 1. = = erpendicular from centre bisects the chord. 2. = = Same as above. 3. = and = In s and 4. = 3. = (given). 4. equal chords of a circle are equidistant from the centre. (Theorem 14.3) 5. = 5. Each = = 6. ommon R.H.S. axiom of congruency. 8. = 8. c.p.c.t = + =. 10. = =. Hence (i) = and (ii) =..E.. 9. From 3 and 8, adding. 10. From 3 and 8, subtracting. Example 10. f two unequal chords of a circle, prove that greater chord is nearer to the centre of the circle. Given. and are chords of a circle with centre. >, and,. To prove. <. onstruction. Join and Understanding ISE mathematics Ix
11 roof. 1. = erpendicular from centre bisects the chord. 2. = Same as above. 3. > 3. > (given) = In, = = In, = = = ( 2 2 ) < 0 2 < 2 <..E.. 6. =, radii of same circle. 7. From 3, > 2 > is +ve. Example 11. If two circles intersect in two points, then prove that the line through their centres is perpendicular bisector of the common chord. Given. Two circles with centres,, and intersecting at points, so that is their common chord. To prove. is perpendicular bisector of i.e. = and = 90. onstruction. Join,, and. roof. In s and 1. = 1. Radii of same circle. 2. = 2. Radii of same circle. 3. = 3. ommon S.S.S. axiom of congruency. 5. = In s and 5. c.p.c.t.. 6. = 6. From = 7. Radii of same circle. 8. = 8. ommon S..S. axiom of congruency. 10. = and = 10. c.p.c.t = is a st. line 12. = 90 Hence, is perpendicular bisector of..e From 10 and 11. circle 1287
12 Exercise alculate the length of a chord which is at a distance 12 cm from the centre of a circle of radius 13 cm. 2. chord of length 48 cm is drawn in a circle of radius 25 cm. alculate its distance from the centre of the circle. 3. chord of length 8 cm is at a distance 3 cm from the centre of the circle. alculate the radius of the circle. 4. alculate the length of a chord which is at a distance 6 cm from the centre of a circle of diameter 20 cm. 5. chord of length 16 cm is at a distance 6 cm from the centre of the circle. Find the length of the chord of the same circle which is at a distance 8 cm from the centre. 6. In a circle of radius 5 cm, and are two parallel chords of length 8 cm and 6 cm respectively. alculate the distance between the chords, if they are on (i) the same side of the centre. (ii) the opposite sides of the centre. 7. (a) In the figure (i) given below, is the centre of the circle. and are two chords of the circle. is perpendicular to and is perpendicular to. = 24 cm, = 5 cm, = 12 cm. Find the: (i) radius of the circle (ii) length of chord. (b) In the figure (ii) given below, is a diameter which meets the chord in E such that E = E = 4 cm. If E = 3 cm, find the radius of the circle. 4 cm 4 cm E 3 cm (i) (ii) 8. In the adjoining figure, and are two parallel chords and is the centre. If the radius of the circle is 15 cm, find the distance between the two chords of length 24 cm and 18 cm respectively. 9. and are two parallel chords of a circle of lengths 10 cm and 4 cm respectively. If the chords lie on the same side of the centre and the distance between them is 3 cm, find the diameter of the circle. 10. is an isosceles triangle inscribed in a circle. If = = 12 5 cm and = 24 cm, find the radius of the circle. 11. n equilateral triangle of side 6 cm is inscribed in a circle. Find the radius of the circle. 12. is a diameter of a circle. is a point in such that = 18 cm and = 8 cm. Find the length of the shortest chord through. 13. rectangle with one side of length 4 cm is inscribed in a circle of diameter 5 cm. Find the area of the rectangle. 14. The length of the common chord of two intersecting circles is 30 cm. If the radii of the two circles are 25 cm and 17 cm, find the distance between their centres Understanding ISE mathematics Ix
Plane geometry Circles: Problems with some Solutions
The University of Western ustralia SHL F MTHMTIS & STTISTIS UW MY FR YUNG MTHMTIINS Plane geometry ircles: Problems with some Solutions 1. Prove that for any triangle, the perpendicular bisectors of the
More informationTheorem 1.2 (Converse of Pythagoras theorem). If the lengths of the sides of ABC satisfy a 2 + b 2 = c 2, then the triangle has a right angle at C.
hapter 1 Some asic Theorems 1.1 The ythagorean Theorem Theorem 1.1 (ythagoras). The lengths a b < c of the sides of a right triangle satisfy the relation a + b = c. roof. b a a 3 b b 4 b a b 4 1 a a 3
More informationCirclesTangent Properties
15 irclestangent roperties onstruction of tangent at a point on the circle. onstruction of tangents when the angle between radii is given. Tangents from an external point  construction and proof Touching
More informationChapter 1. Some Basic Theorems. 1.1 The Pythagorean Theorem
hapter 1 Some asic Theorems 1.1 The ythagorean Theorem Theorem 1.1 (ythagoras). The lengths a b < c of the sides of a right triangle satisfy the relation a 2 + b 2 = c 2. roof. b a a 3 2 b 2 b 4 b a b
More informationProve that a + b = x + y. Join BD. In ABD, we have AOB = 180º AOB = 180º ( 1 + 2) AOB = 180º A
bhilasha lasses lass IX ate: 03 7 SLUTIN (hap 8,9,0) 50 ob no.947967444. The sides and of a quadrilateral are produced as shown in fig. rove that a + b = x + y. Join. In, we have y a + + = 80º = 80º
More informationCircles. Exercise 9.1
9 uestion. Exercise 9. How many tangents can a circle have? Solution For every point of a circle, we can draw a tangent. Therefore, infinite tangents can be drawn. uestion. Fill in the blanks. (i) tangent
More informationChapter (Circle) * Circle  circle is locus of such points which are at equidistant from a fixed point in
Chapter  10 (Circle) Key Concept * Circle  circle is locus of such points which are at equidistant from a fixed point in a plane. * Concentric circle  Circle having same centre called concentric circle.
More informationChapter 19 Exercise 19.1
hapter 9 xercise 9... (i) n axiom is a statement that is accepted but cannot be proven, e.g. x + 0 = x. (ii) statement that can be proven logically: for example, ythagoras Theorem. (iii) The logical steps
More informationTheorems on Area. Introduction Axioms of Area. Congruence area axiom. Addition area axiom
3 Theorems on rea Introduction We know that Geometry originated from the need of measuring land or recasting/refixing its boundaries in the process of distribution of certain land or field among different
More informationMT  GEOMETRY  SEMI PRELIM  II : PAPER  4
017 1100 MT.1. ttempt NY FIVE of the following : (i) In STR, line l side TR S SQ T = RQ x 4.5 = 1.3 3.9 x = MT  GEOMETRY  SEMI RELIM  II : ER  4 Time : Hours Model nswer aper Max. Marks : 40 4.5 1.3
More informationSolve problems involving tangents to a circle. Solve problems involving chords of a circle
8UNIT ircle Geometry What You ll Learn How to Solve problems involving tangents to a circle Solve problems involving chords of a circle Solve problems involving the measures of angles in a circle Why Is
More informationMT EDUCARE LTD. MATHEMATICS SUBJECT : Q L M ICSE X. Geometry STEP UP ANSWERSHEET
IS X MT UR LT. SUJT : MTHMTIS Geometry ST U NSWRSHT 003 1. In QL and RM, LQ MR [Given] LQ RM [Given] QL ~ RM [y axiom of similarity] (i) Since, QL ~ RM QL M L RM QL RM L M (ii) In QL and RQ, we have Q
More informationradii: AP, PR, PB diameter: AB chords: AB, CD, AF secant: AG or AG tangent: semicircles: ACB, ARB minor arcs: AC, AR, RD, BC,
h 6 Note Sheets L Shortened Key Note Sheets hapter 6: iscovering and roving ircle roperties eview: ircles Vocabulary If you are having problems recalling the vocabulary, look back at your notes for Lesson
More informationWhat is the longest chord?.
Section: 76 Topic: ircles and rcs Standard: 7 & 21 ircle Naming a ircle Name: lass: Geometry 1 Period: Date: In a plane, a circle is equidistant from a given point called the. circle is named by its.
More informationMAHESH TUTORIALS. Time : 1 hr. 15 min. Q.1. Solve the following : 3
S.S.. MHESH TUTRILS Test  II atch : S Marks : 30 Date : GEMETRY hapter : 1,, 3 Time : 1 hr. 15 min..1. Solve the following : 3 The areas of two similar triangles are 18 cm and 3 cm respectively. What
More informationCircleChord properties
14 irclehord properties onstruction of a chord of given length. Equal chords are equidistant from the centre. ngles in a segment. ongrue nt circles and concentric circles. onstruction of congruent and
More informationCircles. II. Radius  a segment with one endpoint the center of a circle and the other endpoint on the circle.
Circles Circles and Basic Terminology I. Circle  the set of all points in a plane that are a given distance from a given point (called the center) in the plane. Circles are named by their center. II.
More informationCircles in Neutral Geometry
Everything we do in this set of notes is Neutral. Definitions: 10.1  Circles in Neutral Geometry circle is the set of points in a plane which lie at a positive, fixed distance r from some fixed point.
More informationCHAPTER 10 CIRCLES Introduction
168 MATHEMATICS CIRCLES CHAPTER 10 10.1 Introduction You may have come across many objects in daily life, which are round in shape, such as wheels of a vehicle, bangles, dials of many clocks, coins of
More informationReview for Grade 9 Math Exam  Unit 8  Circle Geometry
Name: Review for Grade 9 Math Exam  Unit 8  ircle Geometry Date: Multiple hoice Identify the choice that best completes the statement or answers the question. 1. is the centre of this circle and point
More informationChords and Arcs. Objectives To use congruent chords, arcs, and central angles To use perpendicular bisectors to chords
 hords and rcs ommon ore State Standards G.. Identify and describe relationships among inscribed angles, radii, and chords. M, M bjectives To use congruent chords, arcs, and central angles To use perpendicular
More informationNew Jersey Center for Teaching and Learning. Progressive Mathematics Initiative
Slide 1 / 150 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the noncommercial use of students
More information10. Circles. Q 5 O is the centre of a circle of radius 5 cm. OP AB and OQ CD, AB CD, AB = 6 cm and CD = 8 cm. Determine PQ. Marks (2) Marks (2)
10. Circles Q 1 True or False: It is possible to draw two circles passing through three given noncollinear points. Mark (1) Q 2 State the following statement as true or false. Give reasons also.the perpendicular
More informationTriangles. Exercise 4.1
4 Question. xercise 4. Fill in the blanks using the correct word given in brackets. (i) ll circles are....(congruent, similar) (ii) ll squares are....(similar, congruent) (iii) ll... triangles are similar.
More informationSSC EXAMINATION GEOMETRY (SETA)
GRND TEST SS EXMINTION GEOMETRY (SET) SOLUTION Q. Solve any five subquestions: [5M] ns. ns. 60 & D have equal height ( ) ( D) D D ( ) ( D) Slope of the line ns. 60 cos D [/M] [/M] tan tan 60 cos cos
More informationRiding a Ferris Wheel. Students should be able to answer these questions after Lesson 10.1:
.1 Riding a Ferris Wheel Introduction to ircles Students should be able to answer these questions after Lesson.1: What are the parts of a circle? How are the parts of a circle drawn? Read Question 1 and
More informationRiding a Ferris Wheel
Lesson.1 Skills Practice Name ate iding a Ferris Wheel Introduction to ircles Vocabulary Identify an instance of each term in the diagram. 1. center of the circle 6. central angle T H I 2. chord 7. inscribed
More informationGeometry: A Complete Course
eometry: omplete ourse with rigonometry) odule  tudent Worket Written by: homas. lark Larry. ollins 4/2010 or ercises 20 22, use the diagram below. 20. ssume is a rectangle. a) f is 6, find. b) f is,
More informationCIRCLES MODULE  3 OBJECTIVES EXPECTED BACKGROUND KNOWLEDGE. Circles. Geometry. Notes
Circles MODULE  3 15 CIRCLES You are already familiar with geometrical figures such as a line segment, an angle, a triangle, a quadrilateral and a circle. Common examples of a circle are a wheel, a bangle,
More informationCommon Core Readiness Assessment 4
ommon ore Readiness ssessment 4 1. Use the diagram and the information given to complete the missing element of the twocolumn proof. 2. Use the diagram and the information given to complete the missing
More informationC Given that angle BDC = 78 0 and DCA = Find angles BAC and DBA.
UNERSTNING IRLE THEREMSPRT NE. ommon terms: (a) R ny portion of a circumference of a circle. (b) HR line that crosses a circle from one point to another. If this chord passes through the centre then
More informationCBSE Class IX Syllabus. Mathematics Class 9 Syllabus
Mathematics Class 9 Syllabus Course Structure First Term Units Unit Marks I Number System 17 II Algebra 25 III Geometry 37 IV Coordinate Geometry 6 V Mensuration 5 Total 90 Second Term Units Unit Marks
More informationIndicate whether the statement is true or false.
PRACTICE EXAM IV Sections 6.1, 6.2, 8.1 8.4 Indicate whether the statement is true or false. 1. For a circle, the constant ratio of the circumference C to length of diameter d is represented by the number.
More informationCOURSE STRUCTURE CLASS IX
environment, observance of small family norms, removal of social barriers, elimination of gender biases; mathematical softwares. its beautiful structures and patterns, etc. COURSE STRUCTURE CLASS IX Units
More informationMathematics Class 9 Syllabus. Course Structure. I Number System 17 II Algebra 25 III Geometry 37 IV Coordinate Geometry 6 V Mensuration 5 Total 90
Mathematics Class 9 Syllabus Course Structure First Term Units Unit Marks I Number System 17 II Algebra 25 III Geometry 37 IV Coordinate Geometry 6 V Mensuration 5 Total 90 Second Term Units Unit Marks
More informationIntermediate Math Circles Wednesday October Problem Set 3
The CETRE for EDUCTI in MTHEMTICS and CMPUTIG Intermediate Math Circles Wednesday ctober 24 2012 Problem Set 3.. Unless otherwise stated, any point labelled is assumed to represent the centre of the circle.
More informationC=2πr C=πd. Chapter 10 Circles Circles and Circumference. Circumference: the distance around the circle
10.1 Circles and Circumference Chapter 10 Circles Circle the locus or set of all points in a plane that are A equidistant from a given point, called the center When naming a circle you always name it by
More informationMT  GEOMETRY  SEMI PRELIM  II : PAPER  5
017 1100 MT MT  GEOMETRY  SEMI PRELIM  II : PPER  5 Time : Hours Model nswer Paper Max. Marks : 40.1. ttempt NY FIVE of the following : (i) X In XYZ, ray YM bisects XYZ XY YZ XM MZ Y Z [Property of
More informationWhat You ll Learn. Why It s Important. We see circles in nature and in design. What do you already know about circles?
We see circles in nature and in design. What do you already know about circles? What You ll Learn ircle properties that relate: a tangent to a circle and the radius of the circle a chord in a circle, its
More informationLesson 1.7 circles.notebook. September 19, Geometry Agenda:
Geometry genda: Warmup 1.6(need to print of and make a word document) ircle Notes 1.7 Take Quiz if you were not in class on Friday Remember we are on 1.7 p.72 not lesson 1.8 1 Warm up 1.6 For Exercises
More informationReady To Go On? Skills Intervention 111 Lines That Intersect Circles
Name ate lass STION 11 Ready To Go On? Skills Intervention 111 Lines That Intersect ircles ind these vocabulary words in Lesson 111 and the Multilingual Glossary. Vocabulary interior of a circle exterior
More informationSM2H Unit 6 Circle Notes
Name: Period: SM2H Unit 6 Circle Notes 6.1 Circle Vocabulary, Arc and Angle Measures Circle: All points in a plane that are the same distance from a given point, called the center of the circle. Chord:
More informationAssignment. Riding a Ferris Wheel Introduction to Circles. 1. For each term, name all of the components of circle Y that are examples of the term.
ssignment ssignment for Lesson.1 Name Date Riding a Ferris Wheel Introduction to ircles 1. For each term, name all of the components of circle Y that are examples of the term. G R Y O T M a. hord GM, R,
More informationMth 076: Applied Geometry (Individualized Sections) MODULE FOUR STUDY GUIDE
Mth 076: pplied Geometry (Individualized Sections) MODULE FOUR STUDY GUIDE INTRODUTION TO GEOMETRY Pick up Geometric Formula Sheet (This sheet may be used while testing) ssignment Eleven: Problems Involving
More informationLesson 12.1 Skills Practice
Lesson 12.1 Skills Practice Introduction to ircles ircle, Radius, and iameter Vocabulary efine each term in your own words. 1. circle circle is a collection of points on the same plane equidistant from
More informationGeo  CH11 Practice Test
Geo  H11 Practice Test Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Identify the secant that intersects ñ. a. c. b. l d. 2. satellite rotates 50 miles
More information= Find the value of n.
nswers: (0 HKM Heat Events) reated by: Mr. Francis Hung Last updated: pril 0 09 099 00  Individual 9 0 0900  Group 0 0 9 0 0 Individual Events I How many pairs of distinct integers between and 0 inclusively
More informationMathematics. Sample Question Paper. Class 10th. (Detailed Solutions) Sample Question Paper 13
Sample uestion aper Sample uestion aper (etailed Solutions) Mathematics lass 0th 5. We have, k and 5 k as three consecutive terms of an. 8 Their common difference will be same. 5 i.e. k k k k 5 k 8 k 8
More informationFill in the blanks Chapter 10 Circles Exercise 10.1 Question 1: (i) The centre of a circle lies in of the circle. (exterior/ interior) (ii) A point, whose distance from the centre of a circle is greater
More informationMT  MATHEMATICS (71) GEOMETRY  PRELIM II  PAPER  6 (E)
04 00 Seat No. MT  MTHEMTIS (7) GEOMETRY  PRELIM II  (E) Time : Hours (Pages 3) Max. Marks : 40 Note : ll questions are compulsory. Use of calculator is not allowed. Q.. Solve NY FIVE of the following
More informationObjectives To find the measure of an inscribed angle To find the measure of an angle formed by a tangent and a chord
13 Inscribed ngles ommon ore State Standards G.. Identify and describe relationships among inscribed angles, radii, and chords. lso G..3, G..4 M 1, M 3, M 4, M 6 bjectives To find the measure of an
More informationCOURSE STRUCTURE CLASS IX Maths
COURSE STRUCTURE CLASS IX Maths Units Unit Name Marks I NUMBER SYSTEMS 08 II ALGEBRA 17 III COORDINATE GEOMETRY 04 IV GEOMETRY 28 V MENSURATION 13 VI STATISTICS & PROBABILITY 10 Total 80 UNIT I: NUMBER
More informationUNIT 6. BELL WORK: Draw 3 different sized circles, 1 must be at LEAST 15cm across! Cut out each circle. The Circle
UNIT 6 BELL WORK: Draw 3 different sized circles, 1 must be at LEAST 15cm across! Cut out each circle The Circle 1 Questions How are perimeter and area related? How are the areas of polygons and circles
More informationMath & 8.7 Circle Properties 8.6 #1 AND #2 TANGENTS AND CHORDS
Math 9 8.6 & 8.7 Circle Properties 8.6 #1 AND #2 TANGENTS AND CHORDS Property #1 Tangent Line A line that touches a circle only once is called a line. Tangent lines always meet the radius of a circle at
More information1. Draw and label a diagram to illustrate the property of a tangent to a circle.
Master 8.17 Extra Practice 1 Lesson 8.1 Properties of Tangents to a Circle 1. Draw and label a diagram to illustrate the property of a tangent to a circle. 2. Point O is the centre of the circle. Points
More informationREVISED vide circular No.63 on
Circular no. 63 COURSE STRUCTURE (FIRST TERM) CLASS IX First Term Marks: 90 REVISED vide circular No.63 on 22.09.2015 UNIT I: NUMBER SYSTEMS 1. REAL NUMBERS (18 Periods) 1. Review of representation of
More informationDefinitions. (V.1). A magnitude is a part of a magnitude, the less of the greater, when it measures
hapter 8 Euclid s Elements ooks V 8.1 V.13 efinitions. (V.1). magnitude is a part of a magnitude, the less of the greater, when it measures the greater. (V.2). The greater is a multiple of the less when
More informationChapterwise questions
hapterwise questions ircles 1. In the given figure, is circumscribing a circle. ind the length of. 3 15cm 5 2. In the given figure, is the center and. ind the radius of the circle if = 18 cm and = 3cm
More informationAREAS RELATED TO CIRCLES
HPTER 1 Points to Remember : RES RELTE T IRLES 1. circle is a collection of points which moves in a plane in such a way that its distance from a fixed point always remains the same. The fixed point is
More informationMT  MATHEMATICS (71) GEOMETRY  PRELIM II  PAPER  1 (E)
04 00 eat No. MT  MTHEMTI (7) GEOMETY  PELIM II  PPE  (E) Time : Hours (Pages 3) Max. Marks : 40 Note : (i) ll questions are compulsory. Use of calculator is not allowed. Q.. olve NY FIVE of the following
More informationTERMWISE SYLLABUS SESSION CLASSIX SUBJECT : MATHEMATICS. Course Structure. Schedule for Periodic Assessments and CASExam. of Session
TERMWISE SYLLABUS SESSION201819 CLASSIX SUBJECT : MATHEMATICS Course Structure Units Unit Name Marks I NUMBER SYSTEMS 08 II ALGEBRA 17 III COORDINATE GEOMETRY 04 IV GEOMETRY 28 V MENSURATION 13 VI STATISTICS
More informationGEOMETRY. Similar Triangles
GOMTRY Similar Triangles SIMILR TRINGLS N THIR PROPRTIS efinition Two triangles are said to be similar if: (i) Their corresponding angles are equal, and (ii) Their corresponding sides are proportional.
More informationAnswers. Chapter10 A Start Thinking. and 4 2. Sample answer: no; It does not pass through the center.
hapter10 10.1 Start Thinking 6. no; is not a right triangle because the side lengths do not satisf the Pthagorean Theorem (Thm. 9.1). 1. (3, ) 7. es; is a right triangle because the side lengths satisf
More informationUNIT OBJECTIVES. unit 9 CIRCLES 259
UNIT 9 ircles Look around whatever room you are in and notice all the circular shapes. Perhaps you see a clock with a circular face, the rim of a cup or glass, or the top of a fishbowl. ircles have perfect
More informationMATHEMATICS (IXX) (CODE NO. 041) Session
MATHEMATICS (IXX) (CODE NO. 041) Session 201819 The Syllabus in the subject of Mathematics has undergone changes from time to time in accordance with growth of the subject and emerging needs of the society.
More informationLabel carefully each of the following:
Label carefully each of the following: Circle Geometry labelling activity radius arc diameter centre chord sector major segment tangent circumference minor segment Board of Studies 1 These are the terms
More informationInequalities for Triangles and Pointwise Characterizations
Inequalities for Triangles and Pointwise haracterizations Theorem (The Scalene Inequality): If one side of a triangle has greater length than another side, then the angle opposite the longer side has the
More informationClick on a topic to go to that section. Euclid defined a circle and its center in this way: Euclid defined figures in this way:
lide 1 / 59 lide / 59 New Jersey enter for eaching and Learning Progressive Mathematics Initiative his material is made freely available at www.njctl.org and is intended for the noncommercial use of students
More informationEXTRA HOTS SUMS CHAPTER : 1  SIMILARITY. = (5 marks) Proof : In ABQ, A ray BP bisects ABQ [Given] AP PQ = AB. By property of an angle...
MT EDURE LTD. EXTR HOTS SUMS HTER : 1  SIMILRITY GEOMETRY 1. isectors of and in meet each other at. Line cuts the side at Q. Then prove that : + Q roof : In Q, ray bisects Q [Given] Q y property of an
More informationAREA RELATED TO CIRCLES
CHAPTER 11 AREA RELATED TO CIRCLES (A) Main Concepts and Results Perimeters and areas of simple closed figures. Circumference and area of a circle. Area of a circular path (i.e., ring). Sector of a circle
More informationChapter 10. Properties of Circles
Chapter 10 Properties of Circles 10.1 Use Properties of Tangents Objective: Use properties of a tangent to a circle. Essential Question: how can you verify that a segment is tangent to a circle? Terminology:
More informationStudy Guide. Exploring Circles. Example: Refer to S for Exercises 1 6.
9 1 Eploring ircles A circle is the set of all points in a plane that are a given distance from a given point in the plane called the center. Various parts of a circle are labeled in the figure at the
More informationClass IX  NCERT Maths Exercise (10.1)
Class IX  NCERT Maths Exercise (10.1) Question 1: Fill in the blanks (i) The centre of a circle lies in of the circle. (exterior/interior) (ii) A point, whose distance from the centre of a circle is greater
More informationSTUDY MATERIAL SUBJECT: MATHEMATICS CLASS  IX
STUDY MATERIAL SUBJECT: MATHEMATICS CLASS  IX 2 INDEX PART  I SA  1 1. Number System 2. Polynomials 3. Coordinate Geometry 4. Introduction to Euclid Geometry 5. Lines and Angles 6. Triangles 7. Heron's
More informationMATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Summative Assessment II. Revision CLASS X Prepared by
MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Summative Assessment II Revision CLASS X 06 7 Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.), B. Ed. Kendriya Vidyalaya GaCHiBOWli
More informationCBSE OSWAAL BOOKS LEARNING MADE SIMPLE. Published by : 1/11, Sahitya Kunj, M.G. Road, Agra , UP (India) Ph.: ,
OSWAAL BOOKS LEARNING MADE SIMPLE CBSE SOLVED PAPER 2018 MATHEMATICS CLASS 9 Published by : OSWAAL BOOKS 1/11, Sahitya Kunj, M.G. Road, Agra  282002, UP (India) Ph.: 0562 2857671, 2527781 email: contact@oswaalbooks.com
More information101 Study Guide and Intervention
opyright Glencoe/McGrawHill, a division of he McGrawHill ompanies, Inc. NM I 101 tudy Guide and Intervention ircles and ircumference arts of ircles circle consists of all points in a plane that are
More informationExercise. and 13x. We know that, sum of angles of a quadrilateral = x = 360 x = (Common in both triangles) and AC = BD
9 Exercise 9.1 Question 1. The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral. Solution Given, the ratio of the angles of quadrilateral are 3 : 5 : 9
More informationSecondary School Certificate Examination Syllabus MATHEMATICS. Class X examination in 2011 and onwards. SSC PartII (Class X)
Secondary School Certificate Examination Syllabus MATHEMATICS Class X examination in 2011 and onwards SSC PartII (Class X) 15. Algebraic Manipulation: 15.1.1 Find highest common factor (H.C.F) and least
More informationMAHESH TUTORIALS. GEOMETRY Chapter : 1, 2, 6. Time : 1 hr. 15 min. Q.1. Solve the following : 3
S.S.C. Test  III Batch : SB Marks : 0 Date : MHESH TUTORILS GEOMETRY Chapter : 1,, 6 Time : 1 hr. 15 min..1. Solve the following : (i) The dimensions of a cuboid are 5 cm, 4 cm and cm. Find its volume.
More informationCalculate the circumference of a circle with radius 5 cm. Calculate the area of a circle with diameter 20 cm.
ETIE F CICLE evision. The terms Diameter, adius, Circumference, rea of a circle should be revised along with the revision of circumference and area. ome straightforward examples should be gone over with
More informationUsing Chords. Essential Question What are two ways to determine when a chord is a diameter of a circle?
10.3 Using hords ssential uestion What are two ways to determine when a chord is a diameter of a circle? rawing iameters OOKI O UU o be proficient in math, you need to look closely to discern a pattern
More information(A) 50 (B) 40 (C) 90 (D) 75. Circles. Circles <1M> 1.It is possible to draw a circle which passes through three collinear points (T/F)
Circles 1.It is possible to draw a circle which passes through three collinear points (T/F) 2.The perpendicular bisector of two chords intersect at centre of circle (T/F) 3.If two arcs of a circle
More information21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle.
21. Prove that If one side of the cyclic quadrilateral is produced then the exterior angle is equal to the interior opposite angle. 22. Prove that If two sides of a cyclic quadrilateral are parallel, then
More informationExercise 10.1 Question 1: Fill in the blanks (i) The centre of a circle lies in of the circle. (exterior/ interior)
Exercise 10.1 Question 1: Fill in the blanks (i) The centre of a circle lies in of the circle. (exterior/ interior) (ii) A point, whose distance from the centre of a circle is greater than its radius lies
More information10.1 Tangents to Circles. Geometry Mrs. Spitz Spring 2005
10.1 Tangents to Circles Geometry Mrs. Spitz Spring 2005 Objectives/Assignment Identify segments and lines related to circles. Use properties of a tangent to a circle. Assignment: Chapter 10 Definitions
More informationAlgebra II/Geometry Curriculum Guide Dunmore School District Dunmore, PA
Algebra II/Geometry Dunmore School District Dunmore, PA Algebra II/Geometry Prerequisite: Successful completion of Algebra 1 Part 2 K Algebra II/Geometry is intended for students who have successfully
More informationEXERCISE 10.1 EXERCISE 10.2
NCERT Class 9 Solved Questions for Chapter: Circle 10 NCERT 10 Class CIRCLES 9 Solved Questions for Chapter: Circle EXERCISE 10.1 Q.1. Fill in the blanks : (i) The centre of a circle lies in of the circle.
More informationCLASS IX GEOMETRY MOCK TEST PAPER
Total time:3hrs darsha vidyalay hunashyal P. M.M=80 STION 10 1=10 1) Name the point in a triangle that touches all sides of given triangle. Write its symbol of representation. 2) Where is thocenter of
More informationIntegrated Math II. IM2.1.2 Interpret given situations as functions in graphs, formulas, and words.
Standard 1: Algebra and Functions Students graph linear inequalities in two variables and quadratics. They model data with linear equations. IM2.1.1 Graph a linear inequality in two variables. IM2.1.2
More informationCongruence. Chapter The Three Points Theorem
Chapter 2 Congruence In this chapter we use isometries to study congruence. We shall prove the Fundamental Theorem of Transformational Plane Geometry, which states that two plane figures are congruent
More informationQUESTION BANK ON STRAIGHT LINE AND CIRCLE
QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,
More informationTHEOREM 10.3 B C In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
10.3 Your Notes pply Properties of hords oal p Use relationships of arcs and chords in a circle. HOM 10.3 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their
More informationGrade 9 Circles. Answer the questions. For more such worksheets visit
ID : ae9circles [1] Grade 9 Circles For more such worksheets visit www.edugain.com Answer the questions (1) Two circles with centres O and O intersect at two points A and B. A line PQ is drawn parallel
More information3. MATHEMATICS (CODE NO. 041) The Syllabus in the subject of Mathematics has undergone changes from time to time in accordance with growth of the subject and emerging needs of the society. The present
More informationChapter  7. (Triangles) Triangle  A closed figure formed by three intersecting lines is called a triangle. A
Chapter  7 (Triangles) Triangle  A closed figure formed by three intersecting lines is called a triangle. A triangle has three sides, three angles and three vertices. Congruent figures  Congruent means
More information10.3 Start Thinking Warm Up Cumulative Review Warm Up
10.3 tart hinking etermine if the statement is always true, sometimes true, or never true. plain your reasoning. 1. chord is a diameter. 2. diameter is a chord. 3. chord and a radius have the same measure.
More informationHonors Geometry Circle Investigation  Instructions
Honors Geometry ircle Investigation  Instructions 1. On the first circle a. onnect points and O with a line segment. b. onnect points O and also. c. Measure O. d. Estimate the degree measure of by using
More informationDO NOW #1. Please: Get a circle packet
irclengles.gsp pril 26, 2013 Please: Get a circle packet Reminders: R #10 due Friday Quiz Monday 4/29 Quiz Friday 5/3 Quiz Wednesday 5/8 Quiz Friday 5/10 Initial Test Monday 5/13 ctual Test Wednesday 5/15
More informationTRIPURA BOARD OF SECONDARY EDUCATION. SYLLABUS (effective from 2016) SUBJECT : MATHEMATICS (Class IX)
TRIPURA BOARD OF SECONDARY EDUCATION SYLLABUS (effective from 2016) SUBJECT : MATHEMATICS (Class IX) Total Page 10 MATHEMATICS COURSE STRUCTURE Class IX HALF YEARLY One Paper Time: 3 Hours Marks: 90 Unit
More information