Circle theorems CHAPTER 19


 Bertha Fitzgerald
 2 years ago
 Views:
Transcription
1 352 CHTE 19 Circle theorems In this chapter you will learn how to: use correct vocabulary associated with circles use tangent properties to solve problems prove and use various theorems about angle properties inside a circle prove and use the alternate segment (intersecting tangent and chord) theorem prove and use the intersecting chords theorem. You will also be challenged to: investigate the ninepoint circle theorem. Starter: Circle vocabulary Here are some words you will often encounter when working with circles: Centre adius Chord Diameter Circumference Tangent rc Sector Segment Match the correct words to the nine diagrams below: a) The French call a rainbow an arcenciel (ciel sky). Do you think this is a good name? b) The English call a piece of an orange a segment. Do you think this is a good name?
2 19.1 Tangents, chords and circles Tangents, chords and circles In this section you will learn some theorems about circles, and then use them to solve problems. The theorems are concerned with tangents and chords. chord is a line segment joining two points on the circumference of a circle. tangent is a straight line that touches a circle only once. line segment drawn from the centre of a circle to the midpoint of a chord will intersect the chord at right angles. If M is the midpoint of, i.e. M and M are the same length M then these two angles will both be right angles. tangent and radius meet at right angles. adius The fact that the radius and the tangent meet at right angles is very obvious when one is vertical and the other is horizontal Tangent adius but is not quite so obvious when the situation is rotated, like this. Tangent
3 354 Chapter 19: Circle theorems The two eternal tangents to a circle are equal in length. Tangent Tangent These theorems may be used to help you determine the values of missing angles in circles. When you use them, remember to tell the eaminer which theorem(s) you have used. EMLE The diagram shows a circle, centre. T is a tangent to the circle. Find the value of. 48 T SLUTIN ngle T 90 (angle between the radius and tangent is 90 ). The angles in triangle T add up to 180, so:
4 19.1 Tangents, chords and circles 355 EMLE The diagram shows a circle, centre. is a chord across the circle. M is the midpoint of. Find the value of y. SLUTIN ngle M 90 (radius bisecting chord). The angles in triangle M add up to 180. y 62 M So: y y y y 28 Since a radius bisects a chord at right angles, there is often an opportunity to use ythagoras theorem. EMLE The diagram shows a radius T that bisects the chord at M. M 12 cm. The radius of the circle is 13 cm. Work out the length MT. M T SLUTIN First join : Now apply ythagoras theorem to triangle M: M M M 25 5 cm The distance T is a radius, that is, 13 cm. Thus MT cm T
5 356 Chapter 19: Circle theorems EECISE T is a tangent to the circle, centre. ngle T 29. y 29 T a) State, with a reason, the value of the angle marked. b) Work out the value of the angle marked y. 2 T is a tangent to the circle, centre. T 24 cm. 7 cm. 7 cm 24 cm T The line T intersects the circle at, as shown. Work out the length of T. 3 T and T are tangents to the circle, centre. ngle T is T a) Work out the size of angle. Give reasons. b) What type of quadrilateral is T? Eplain your reasoning.
6 19.1 Tangents, chords and circles T and T are tangents to the circle, centre. ngle is y T a) What type of triangle is triangle? b) Work out the value of. c) Work out the value of y. 5 The diagram shows a circle, centre. The radius of the circle is 5 cm. M is the midpoint of EF. M 3 cm. E M F Calculate the length of EF. 6 The diagram shows a circle, centre. 34 cm. M is the midpoint of. M 8 cm. M Work out the radius of the circle.
7 358 Chapter 19: Circle theorems 7 From a point T, two tangents T and T are drawn to a circle, centre. a) Make a sketch to show this information. The length T is measured, and found to be eactly the same as the length. b) What type of quadrilateral is T? 8 The diagram shows a circle, centre. and CD are chords. The radius T passes through the midpoints M and N of the chords. M 8 cm, NT 3 cm, 30 cm. 8 cm M C N 3 cm T D a) Eplain why angle M 90. b) Use ythagoras theorem to calculate the distance. Show your working. c) Calculate the distance MN. d) Calculate the length of the chord CD.
8 19.2 ngle properties inside a circle ngle properties inside a circle There are several important theorems about angles inside a circle. You will need to learn these, and use them to solve numerical problems. You may also be asked to prove why they are true. Consider two points, and say, on the circumference of a circle. The angle subtended by the arc at the centre is angle. ngle subtended by arc at The angle subtended by the arc at a point on the circumference is angle. ngle subtended by arc at There is a theorem in circle geometry which states that angle is eactly twice angle. This angle at the centre is twice as big as this one, at the circumference. 2 This result is quite easy to prove, and is the basic theorem upon which several other circle theorems are built.
9 360 Chapter 19: Circle theorems THEEM The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at the circumference of the circle. F Make a diagram to show the arc, the centre, and the point on the circumference of the circle: From, draw a radius to, and produce it, which means etend it slightly: Triangle is isosceles, since both and are radii of the same circle. Therefore angles and are equal. These are marked on the diagram with a letter a: a a Likewise the triangle is isosceles, since both and are radii of the same circle. Therefore angles and are equal. These are marked on the diagram with a letter b: The angle at the circumference is angle a b. a a b b
10 19.2 ngle properties inside a circle 361 To obtain an epression for the angle at the centre, look at this magnified copy of the diagram: a b 180 2a 180 2b 2a 2b a Y b ngle 180 a a 180 2a So: angle Y 180 (180 2a) 2a Likewise: angle 180 b b 180 2b So: angle Y 180 (180 2b) 2b Thus the angle at the centre is: ngle 2a 2b 2(a b) ut angle a b, from above. Therefore angle 2 angle. Thus, the angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at the circumference of the circle. Two further theorems can be deduced immediately from this first one. You can quote the previous theorem to justify these proofs.
11 362 Chapter 19: Circle theorems THEEM ngles subtended by an arc in the same segment of a circle are equal. These three angles are all equal. F Join and so that they form an angle at the centre: If the angle at is, then the angle at the centre must be 2 2 and if the angle at the centre is 2, then the angle at must be. Thus angles and are equal. THEEM The angle subtended in a semicircle is a right angle. F Since is a diameter, is a straight line. If this line is a diameter Thus angle 180. Using the result that the angle at the circumference is half that at the centre: ngle then this angle will be a right angle.
12 19.2 ngle properties inside a circle 363 EMLE Find the values of the angles marked and y. Eplain your reasoning in both cases Diagram not to scale y SLUTIN ngle 44 (angles in the same segment are equal). 70 y 180 (angles in a triangle add up to 180 ) and 44, so: y y 180 y y 66 EMLE Find the values of the angles marked and y. Eplain your reasoning in both cases y SLUTIN y (angle at centre 2 angle at circumference) (angle in a semicircle is a right angle and angles in a triangle add up to 180 )
13 364 Chapter 19: Circle theorems EECISE 19.2 Find the missing angles in these diagrams, which are not drawn to scale. Eplain your reasoning in each case d 48 a f g 28 b 44 c e h 78 i j k 104 l m n p s 36 q r t u 75 v 2v
14 19.3 Further circle theorems Further circle theorems THEEM The angles subtended in opposite segments add up to 180. This angle plus this one add up to 180. F Denote the angles and as p and q respectively. p Then the angles at the centre are twice these, that is, 2p and 2q. ngles at point add up to 360, so: 2p 2q 360 2q 2p Thus 2(p q) 360 So p q 180 q The points,, and form a quadrilateral whose vertices lie around a circle; it is known as a cyclic quadrilateral. Thus the theorem may also be stated as: pposite angles of a cyclic quadrilateral add up to 180
15 366 Chapter 19: Circle theorems EMLE Find the angles and y. Diagram not to scale y SLUTIN For angle, we have (angles in opposite segments) Thus For angle y, two construction lines are needed: a y 96 Use angles in opposite segments, a Using the angle at centre is twice the angle at circumference: y
16 19.3 Further circle theorems 367 THEEM The angle between a tangent and chord is equal to the angle subtended in the opposite segment. (This is often called the alternate segment theorem.) This angle is equal to this one. F First, consider the special case of a tangent meeting a diameter: y Since the angle in a semicircle is 90, the other two angles in the triangle add up to 90. Hence y z 90. Since a radius and tangent meet at 90 : z 90 Hence z y z. From which it follows that y. z Now move around the circle to, say, so that it is no longer on the end of a diameter. The angle at is equal to the angle at, as they are angles in the same segment. Thus the theorem is proved. y y z
17 368 Chapter 19: Circle theorems EECISE 19.3 Find the missing angles in these diagrams, which are not drawn to scale. Eplain your reasoning in each case a b f d c e g h i j k l n m q s 87 p r t 52 v w 132 u
18 19.4 Intersecting chords Intersecting chords THEEM Consider a pair of chords and CD intersecting at a point inside a circle, as shown in this diagram: C D Then the lengths,, C and D are related by the following result: C D F Join C and D to complete two triangles C and D as below: C Then angles C and D are equal (angles in the same segment). ngles C and D are equal (angles in the same segment). ngles C and D are equal (vertically opposite). Thus the triangles C and D are mathematically similar, so triangle D is an enlargement of triangle C. The enlargement factor is C, but it is also D, and these must be equal, so C D Crossmultiplying, C D and the result is proved. D
19 370 Chapter 19: Circle theorems EMLE Find the missing length in the diagram below. 5 cm 8 cm 10 cm SLUTIN Using the result for intersecting chords we have So the missing length is 4 cm. S The same principle is valid even when the two chords cross over outside the circle, as in the net eample. EMLE and S are chords of a circle, and, when produced, intersect at. 10 cm 10 cm. S 6 cm. 9 cm. a) Write down the length S. b) Work out the length. c) Work out the length. 6 cm 9 cm SLUTIN a) S cm S b) S cm c) cm
20 19.4 Intersecting chords 371 EECISE Find the missing length, cm, in this diagram. 10 cm S 3 cm 9 cm cm 2 Find the missing length, y cm, in this diagram. y cm 12 cm S 5 cm 10 cm 3 Chords and DC, when produced, meet at point. 3 cm. 12 cm. C 10 cm. 3 cm 12 cm 10 cm cm C D a) Work out the length D. b) Hence work out the length DC, marked as cm. 4 In the diagram below, is a diameter of the circle. 2 cm. 6 cm. S 4 cm. cm. 6 cm 2 cm cm 4 cm S a) Work out the value of. b) Hence work out the radius of the circle.
21 372 Chapter 19: Circle theorems 5 Chords and S, when produced, meet at point W. 7 cm. W 8 cm. SW 10 cm. a) Work out the length W. b) Hence work out the length S, marked as cm. W 8 cm 7 cm 10 cm S cm EVIEW EECISE 19 1 circle of diameter 10 cm has a chord drawn inside it. The chord is 7 cm long. a) Make a sketch to show this information. b) Calculate the distance from the midpoint of the chord to the centre of the circle. Give your answer correct to 3 significant figures. 2 The diagram shows a circle, centre. T and T are tangents to the circle. ngle T a) Work out the size of angle T, marked. b) Is it possible to draw a circle that passes through the four points,, and T? Give reasons for your answer. 3 The diagram shows a circle, centre. T and T are tangents to the circle. ngle T 32. S y 32 T a) Work out the size of angle S, marked y. Hint: Draw in and. b) Is it possible to draw a circle that passes through the four points, S, and T? Give reasons for your answer.
22 eview eercise In the diagram,, and C are points on the circle, centre. ngle CE 63. FE is a tangent to the circle at point C. E Diagram not accurately drawn 63 C F a) Calculate the size of angle C. Give reasons for your answer. b) Calculate the size of angle C. Give reasons for your answer. [Edecel] 5,, and S are points on the circumference of a circle, centre. is a diameter of the circle. ngle S 56. S 56 Diagram not accurately drawn a) Find the size of angle. Give a reason for your answer. b) Find the size of angle. Give a reason for your answer. c) Find the size of angle. Give a reason for your answer. [Edecel]
23 374 Chapter 19: Circle theorems 6,, C and D are four points on the circumference of a circle. E and DCE are straight lines. ngle C 25. ngle EC 60. a) Find the size of angle DC. b) Find the size of angle D. 25 Diagram not accurately drawn 60 D C E ngle CD 65. en says that D is a diameter of the circle. c) Is en correct? You must eplain your answer. [Edecel] 7 The diagram shows a circle, centre. C is a diameter. ngle C 35. D is the point on C such that angle D is a right angle. Diagram not accurately drawn 35 D C a) Work out the size of angle C. Give reasons for your answer. b) Calculate the size of angle DC. c) Calculate the size of angle. [Edecel]
24 eview eercise ,, C and D are four points on the circumference of a circle. T is the tangent to the circle at. ngle DT 30. ngle DC 132. C Diagram not accurately drawn 132 D 30 T a) Calculate the size of angle C. Eplain your method. b) Calculate the size of angle CD. Eplain your method. c) Eplain why C cannot be a diameter of the circle. [Edecel] 9 oints, and C lie on the circumference of a circle with centre. D is the tangent to the circle at. CD is a straight line. C and intersect at E. Diagram not accurately drawn E C D ngle C 80. ngle CD 38. a) Calculate the size of angle C. b) Calculate the size of angle. c) Give a reason why it is not possible to draw a circle with diameter ED through the point. [Edecel]
25 376 Chapter 19: Circle theorems 10,, C and D are points on the circumference of a circle centre. tangent is drawn from E to touch the circle at C. ngle EC 36. E is a straight line. Diagram not accurately drawn E 36 D a) Calculate the size of angle C. Give reasons for your answer. b) Calculate the size of angle DC. Give reasons for your answer. [Edecel] 11, and are points on a circle. is the centre of the circle. T is the tangent to the circle at. ngle T 56. C Diagram not accurately drawn 56 T a) Find (i) the size of angle and (ii) the size of angle.,, C and D are points on a circle. C is a diameter of the circle. ngle CD 25 and angle CD 132. Diagram not accurately drawn C D b) Calculate (i) the size of angle C and (ii) the size of angle D. [Edecel]
26 Key points In the diagram below, and S are chords of the circle. 4 cm. 6 cm. 3 cm. S cm. 3 cm 4 cm 6 cm cm S Work out the value of. Key points asic circle properties line segment drawn from the centre of a circle to the midpoint of a chord will intersect the chord at right angles. tangent and radius meet at right angles. The two eternal tangents to a circle are equal in length. Tangent M adius Tangent Tangent
27 378 Chapter 19: Circle theorems Circle theorems The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at the circumference of the circle. ngles subtended by an arc in the same segment of a circle are equal. The angle subtended in a semicircle is a right angle. 2 The angles subtended in opposite segments add up to 180. y 180 The angle between a tangent and chord is equal to the angle subtended in the opposite segment. y Intersecting chords (internal) Intersecting chords (eternal) C C D D C D C D
28 Internet Challenge Internet Challenge 19 The ninepoint circle theorem This diagram shows the ninepoint circle. Here are the instructions to make it M C Start with any general triangle, whose vertices are, and C. Construct points 1, 2, 3. Can you see what rule is used to locate them? Construct the point M. Can you see how 1, 2, 3 are used to do this? Construct points 4, 5, 6. Can you see what rule is used to locate them? Construct points 7, 8, 9. Can you see what rule is used to locate them? Then it should be possible to draw a circle that passes through all nine of the points: 1, 2, 3, 4, 5, 6, 7, 8 and 9. Look at the diagram, and see if you can figure out how the various points are constructed. Use the internet to check that your deductions are correct. Try to make a ninepoint circle of your own, using compass constructions. You might also try to do this using computer graphics software. Which mathematician is thought to have first made a ninepoint circle? Can you find a proof that these nine points all lie on the same circle?
MATHEMATICS Grade 12 EUCLIDEAN GEOMETRY: CIRCLES 02 JULY 2014
EUCLIDEAN GEOMETRY: CIRCLES 02 JULY 2014 Checklist Make sure you learn proofs of the following theorems: The line drawn from the centre of a circle perpendicular to a chord bisects the chord The angle
More informationInscribed Angle Theorem and Its Applications
: Student Outcomes Prove the inscribed angle theorem: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle. Recognize and use different
More informationEUCLIDEAN GEOMETRY: (±50 marks)
ULIN GMTRY: (±50 marks) Grade theorems:. The line drawn from the centre of a circle perpendicular to a chord bisects the chord. 2. The perpendicular bisector of a chord passes through the centre of the
More informationGeometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: Activity 24
Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: ctivity 24 esources: Springoard Geometry Unit Overview In this unit, students will study formal definitions of basic figures,
More informationRadius, diameter, circumference, π (Pi), central angles, Pythagorean relationship. about CIRCLES
Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 1 Chapter 8 in textbook (p. 384 420) 5/50 or 10% on 2011 CRT: 5 Multiple Choice WHAT YOU SHOULD ALREADY KNOW: Radius, diameter, circumference, π (Pi), central
More informationCircle Name: Radius: Diameter: Chord: Secant:
12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane
More informationLesson 2: Circles, Chords, Diameters, and Their Relationships
Circles, Chords, Diameters, and Their Relationships Student Outcomes Identify the relationships between the diameters of a circle and other chords of the circle. Lesson Notes Students are asked to construct
More informationUnknown Angle Problems with Inscribed Angles in Circles
: Unknown Angle Problems with Inscribed Angles in Circles Student Outcomes Use the inscribed angle theorem to find the measures of unknown angles. Prove relationships between inscribed angles and central
More informationC1: Coordinate geometry of straight lines
B_Chap0_0805.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
More information206 MATHEMATICS CIRCLES
206 MATHEMATICS 10.1 Introduction 10 You have studied in Class IX that a circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre). You have
More informationThe measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures
8.1 Name (print first and last) Per Date: 3/24 due 3/25 8.1 Circles: Arcs and Central Angles Geometry Regents 20132014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to
More informationCircles Learning Strategies. What should students be able to do within this interactive?
Circles Learning Strategies What should students be able to do within this interactive? Select one of the four circle properties. Follow the directions provided for each property. Perform the investigations
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More informationGrade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013
Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is
More informationSenior Math Circles: Geometry I
Universit of Waterloo Facult of Mathematics entre for Education in Mathematics and omputing pening Problem (a) If 30 7 = + + z Senior Math ircles: Geometr I, where, and z are positive integers, then what
More informationGeometry SOL G.11 G.12 Circles Study Guide
Geometry SOL G.11 G.1 Circles Study Guide Name Date Block Circles Review and Study Guide Things to Know Use your notes, homework, checkpoint, and other materials as well as flashcards at quizlet.com (http://quizlet.com/4776937/chapter10circlesflashcardsflashcards/).
More informationGEOMETRIC MENSURATION
GEOMETRI MENSURTION Question 1 (**) 8 cm 6 cm θ 6 cm O The figure above shows a circular sector O, subtending an angle of θ radians at its centre O. The radius of the sector is 6 cm and the length of the
More informationFor the circle above, EOB is a central angle. So is DOE. arc. The (degree) measure of ù DE is the measure of DOE.
efinition: circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. We use the symbol to represent a circle. The a line segment from the center
More informationNCERT. Area of the circular path formed by two concentric circles of radii. Area of the sector of a circle of radius r with central angle θ =
AREA RELATED TO CIRCLES (A) Main Concepts and Results CHAPTER 11 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 informationIsosceles triangles. Key Words: Isosceles triangle, midpoint, median, angle bisectors, perpendicular bisectors
Isosceles triangles Lesson Summary: Students will investigate the properties of isosceles triangles. Angle bisectors, perpendicular bisectors, midpoints, and medians are also examined in this lesson. A
More informationCircle Theorems. This circle shown is described an OT. As always, when we introduce a new topic we have to define the things we wish to talk about.
Circle s circle is a set of points in a plane that are a given distance from a given point, called the center. The center is often used to name the circle. T This circle shown is described an OT. s always,
More informationChapter 6 Notes: Circles
Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment
More informationCK12 Geometry: Parts of Circles and Tangent Lines
CK12 Geometry: Parts of Circles and Tangent Lines Learning Objectives Define circle, center, radius, diameter, chord, tangent, and secant of a circle. Explore the properties of tangent lines and circles.
More informationGeo 9 1 Circles 91 Basic Terms associated with Circles and Spheres. Radius. Chord. Secant. Diameter. Tangent. Point of Tangency.
Geo 9 1 ircles 91 asic Terms associated with ircles and Spheres ircle Given Point = Given distance = Radius hord Secant iameter Tangent Point of Tangenc Sphere Label ccordingl: ongruent circles or spheres
More informationGeometry in a Nutshell
Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where
More informationGEOMETRY. Constructions OBJECTIVE #: G.CO.12
GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic
More informationTangent Properties. Line m is a tangent to circle O. Point T is the point of tangency.
CONDENSED LESSON 6.1 Tangent Properties In this lesson you will Review terms associated with circles Discover how a tangent to a circle and the radius to the point of tangency are related Make a conjecture
More information1 Solution of Homework
Math 3181 Dr. Franz Rothe February 4, 2011 Name: 1 Solution of Homework 10 Problem 1.1 (Common tangents of two circles). How many common tangents do two circles have. Informally draw all different cases,
More informationCalculate the circumference of a circle with radius 5 cm. Calculate the area of a circle with diameter 20 cm.
RERTIES F CIRCLE Revision. The terms Diameter, Radius, Circumference, rea of a circle should be revised along with the revision of circumference and area. Some straightforward examples should be gone over
More informationChapters 6 and 7 Notes: Circles, Locus and Concurrence
Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of
More informationChapter 4 Circles, TangentChord Theorem, Intersecting Chord Theorem and Tangentsecant Theorem
Tampines Junior ollege H3 Mathematics (9810) Plane Geometry hapter 4 ircles, Tangenthord Theorem, Intersecting hord Theorem and Tangentsecant Theorem utline asic definitions and facts on circles The
More informationIMO Training 2008 Circles Yufei Zhao. Circles. Yufei Zhao.
ircles Yufei Zhao yufeiz@mit.edu 1 Warm up problems 1. Let and be two segments, and let lines and meet at X. Let the circumcircles of X and X meet again at O. Prove that triangles O and O are similar.
More informationGeometry Unit 10 Notes Circles. Syllabus Objective: 10.1  The student will differentiate among the terms relating to a circle.
Geometry Unit 0 Notes ircles Syllabus Objective: 0.  The student will differentiate among the terms relating to a circle. ircle the set of all points in a plane that are equidistant from a given point,
More informationcircle the set of all points that are given distance from a given point in a given plane
Geometry Week 19 Sec 9.1 to 9.3 Definitions: section 9.1 circle the set of all points that are given distance from a given point in a given plane E D Notation: F center the given point in the plane radius
More information39 Symmetry of Plane Figures
39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that
More informationTangents and Chords Off On a Tangent
SUGGESTED LERNING STRTEGIES: Group Presentation, Think/Pair/Share, Quickwrite, Interactive Word Wall, Vocabulary Organizer, Create Representations, Quickwrite circle is the set of all points in a plane
More information1. Find the length of BC in the following triangles. It will help to first find the length of the segment marked X.
1 Find the length of BC in the following triangles It will help to first find the length of the segment marked X a: b: Given: the diagonals of parallelogram ABCD meet at point O The altitude OE divides
More informationSection 91. Basic Terms: Tangents, Arcs and Chords Homework Pages 330331: 118
Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,
More informationWarm Up #23: Review of Circles 1.) A central angle of a circle is an angle with its vertex at the of the circle. Example:
Geometr hapter 12 Notes  1  Warm Up #23: Review of ircles 1.) central angle of a circle is an angle with its verte at the of the circle. Eample: X 80 2.) n arc is a section of a circle. Eamples:, 3.)
More informationClass10 th (X) Mathematics Chapter: Tangents to Circles
Class10 th (X) Mathematics Chapter: Tangents to Circles 1. Q. AB is line segment of length 24 cm. C is its midpoint. On AB, AC and BC semicircles are described. Find the radius of the circle which touches
More informationCCGPS UNIT 3 Semester 1 ANALYTIC GEOMETRY Page 1 of 32. Circles and Volumes Name:
GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 1 of 3 ircles and Volumes Name: ate: Understand and apply theorems about circles M91.G..1 Prove that all circles are similar. M91.G.. Identify and describe relationships
More informationAngles in a Circle and Cyclic Quadrilateral
130 Mathematics 19 Angles in a Circle and Cyclic Quadrilateral 19.1 INTRODUCTION You must have measured the angles between two straight lines, let us now study the angles made by arcs and chords in a circle
More informationArchimedes and the Arbelos 1 Bobby Hanson October 17, 2007
rchimedes and the rbelos 1 obby Hanson October 17, 2007 The mathematician s patterns, like the painter s or the poet s must be beautiful; the ideas like the colours or the words, must fit together in a
More informationAdvanced Euclidean Geometry
dvanced Euclidean Geometry What is the center of a triangle? ut what if the triangle is not equilateral?? Circumcenter Equally far from the vertices? P P Points are on the perpendicular bisector of a line
More information1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
More informationDuplicating Segments and Angles
CONDENSED LESSON 3.1 Duplicating Segments and ngles In this lesson, you Learn what it means to create a geometric construction Duplicate a segment by using a straightedge and a compass and by using patty
More informationSMT 2014 Geometry Test Solutions February 15, 2014
SMT 014 Geometry Test Solutions February 15, 014 1. The coordinates of three vertices of a parallelogram are A(1, 1), B(, 4), and C( 5, 1). Compute the area of the parallelogram. Answer: 18 Solution: Note
More informationArc Length and Areas of Sectors
Student Outcomes When students are provided with the angle measure of the arc and the length of the radius of the circle, they understand how to determine the length of an arc and the area of a sector.
More informationThe Inscribed Angle Alternate A Tangent Angle
Student Outcomes Students use the inscribed angle theorem to prove other theorems in its family (different angle and arc configurations and an arc intercepted by an angle at least one of whose rays is
More informationGeometry of 2D Shapes
Name: Geometry of 2D Shapes Answer these questions in your class workbook: 1. Give the definitions of each of the following shapes and draw an example of each one: a) equilateral triangle b) isosceles
More informationCIRCLE COORDINATE GEOMETRY
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
More informationLesson 1: Introducing Circles
IRLES N VOLUME Lesson 1: Introducing ircles ommon ore Georgia Performance Standards M9 12.G..1 M9 12.G..2 Essential Questions 1. Why are all circles similar? 2. What are the relationships among inscribed
More informationTest on Circle Geometry (Chapter 15)
Test on Circle Geometry (Chapter 15) Chord Properties of Circles A chord of a circle is any interval that joins two points on the curve. The largest chord of a circle is its diameter. 1. Chords of equal
More informationAngles that are between parallel lines, but on opposite sides of a transversal.
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
More informationEquation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1
Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows : gradient = vertical horizontal horizontal A B vertical
More informationElementary triangle geometry
Elementary triangle geometry Dennis Westra March 26, 2010 bstract In this short note we discuss some fundamental properties of triangles up to the construction of the Euler line. ontents ngle bisectors
More informationWarmup Tangent circles Angles inside circles Power of a point. Geometry. Circles. Misha Lavrov. ARML Practice 12/08/2013
Circles ARML Practice 12/08/2013 Solutions Warmup problems 1 A circular arc with radius 1 inch is rocking back and forth on a flat table. Describe the path traced out by the tip. 2 A circle of radius
More informationContents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...
Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................
More informationCentroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More informationMaximizing Angle Counts for n Points in a Plane
Maximizing Angle Counts for n Points in a Plane By Brian Heisler A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE OF STETSON UNIVERSITY IN PARTIAL FULFILLMENT OF
More informationMATH STUDENT BOOK. 8th Grade Unit 6
MATH STUDENT BOOK 8th Grade Unit 6 Unit 6 Measurement Math 806 Measurement Introduction 3 1. Angle Measures and Circles 5 Classify and Measure Angles 5 Perpendicular and Parallel Lines, Part 1 12 Perpendicular
More informationUnit 3 Circles and Spheres
Accelerated Mathematics I Frameworks Student Edition Unit 3 Circles and Spheres 2 nd Edition March, 2011 Table of Contents INTRODUCTION:... 3 Sunrise on the First Day of a New Year Learning Task... 8 Is
More informationStudent Name: Teacher: Date: District: MiamiDade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1
Student Name: Teacher: Date: District: MiamiDade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the
More informationTest to see if ΔFEG is a right triangle.
1. Copy the figure shown, and draw the common tangents. If no common tangent exists, state no common tangent. Every tangent drawn to the small circle will intersect the larger circle in two points. Every
More informationGeometry Chapter 9. Circle Vocabulary Arc Length Angle & Segment Theorems with Circles Proofs
Geometry hapter 9 ircle Vocabulary rc Length ngle & Segment Theorems with ircles Proofs hapter 9: ircles Date Due Section Topics ssignment 9.1 9.2 Written Eercises Definitions Worksheet (pg330 classroom
More informationObjective: To distinguish between degree and radian measure, and to solve problems using both.
CHAPTER 3 LESSON 1 Teacher s Guide Radian Measure AW 3.2 MP 4.1 Objective: To distinguish between degree and radian measure, and to solve problems using both. Prerequisites Define the following concepts.
More informationUnit 10 Geometry Circles. NAME Period
Unit 10 Geometry Circles NAME Period 1 Geometry Chapter 10 Circles ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. *** 1. (101) Circles and Circumference
More information4. An isosceles triangle has two sides of length 10 and one of length 12. What is its area?
1 1 2 + 1 3 + 1 5 = 2 The sum of three numbers is 17 The first is 2 times the second The third is 5 more than the second What is the value of the largest of the three numbers? 3 A chemist has 100 cc of
More informationcircumscribed circle Vocabulary Flash Cards Chapter 10 (p. 539) Chapter 10 (p. 530) Chapter 10 (p. 538) Chapter 10 (p. 530)
Vocabulary Flash ards adjacent arcs center of a circle hapter 10 (p. 539) hapter 10 (p. 530) central angle of a circle chord of a circle hapter 10 (p. 538) hapter 10 (p. 530) circle circumscribed angle
More informationCongruence. Set 5: Bisectors, Medians, and Altitudes Instruction. Student Activities Overview and Answer Key
Instruction Goal: To provide opportunities for students to develop concepts and skills related to identifying and constructing angle bisectors, perpendicular bisectors, medians, altitudes, incenters, circumcenters,
More informationGeometry Unit 5: Circles Part 1 Chords, Secants, and Tangents
Geometry Unit 5: Circles Part 1 Chords, Secants, and Tangents Name Chords and Circles: A chord is a segment that joins two points of the circle. A diameter is a chord that contains the center of the circle.
More informationCircles  Past Edexcel Exam Questions
ircles  Past Edecel Eam Questions 1. The points A and B have coordinates (5,1) and (13,11) respectivel. (a) find the coordinates of the midpoint of AB. [2] Given that AB is a diameter of the circle,
More informationCIRCLE DEFINITIONS AND THEOREMS
DEFINITIONS Circle The set of points in a plane equidistant from a given point(the center of the circle). Radius A segment from the center of the circle to a point on the circle(the distance from the
More informationGeometry Chapter 10 Study Guide Name
eometry hapter 10 Study uide Name Terms and Vocabulary: ill in the blank and illustrate. 1. circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center.
More informationTwice the Angle.  Circle Theorems 3: Angle at the Centre Theorem 
 Circle Theorems 3: Angle at the Centre Theorem  Definitions An arc of a circle is a contiguous (i.e. no gaps) portion of the circumference. An arc which is half of a circle is called a semicircle.
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications
More informationCSU Fresno Problem Solving Session. Geometry, 17 March 2012
CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfdprep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news
More informationUnit 3 Practice Test. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: lass: ate: I: Unit 3 Practice Test Multiple hoice Identify the choice that best completes the statement or answers the question. The radius, diameter, or circumference of a circle is given. Find
More informationLesson 6.3: Arcs and Angles
Lesson 6.3: Arcs and Angles In this lesson you will: make conjectures about inscribed angles in a circle investigate relationships among the angles in a cyclic quadrilateral compare the arcs formed when
More informationThe Geometry of Piles of Salt Thinking Deeply About Simple Things
The Geometry of Piles of Salt Thinking Deeply About Simple Things PCMI SSTP Tuesday, July 15 th, 2008 By Troy Jones Willowcreek Middle School Important Terms (the word line may be replaced by the word
More informationUnit 3: Circles and Volume
Unit 3: Circles and Volume This unit investigates the properties of circles and addresses finding the volume of solids. Properties of circles are used to solve problems involving arcs, angles, sectors,
More information104 Inscribed Angles. Find each measure. 1.
Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semicircle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what
More informationChapter Review. 111 Lines that Intersect Circles. 112 Arcs and Chords. Identify each line or segment that intersects each circle.
HPTR 111 hapter Review 111 Lines that Intersect ircles Identify each line or segment that intersects each circle. 1. m 2. N L K J n W Y X Z V 3. The summit of Mt. McKinley in laska is about 20,321 feet
More informationThree Lemmas in Geometry
Winter amp 2010 Three Lemmas in Geometry Yufei Zhao Three Lemmas in Geometry Yufei Zhao Massachusetts Institute of Technology yufei.zhao@gmail.com 1 iameter of incircle T Lemma 1. Let the incircle of triangle
More informationStudent Name: Teacher: Date: District: MiamiDade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 3. Form: 201
Student Name: Teacher: District: Date: MiamiDade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 3 Description: Geometry Topic 6: Circles Form: 201 1. Which method is valid for proving
More informationGeometry: Euclidean. Through a given external point there is at most one line parallel to a
Geometry: Euclidean MATH 3120, Spring 2016 The proofs of theorems below can be proven using the SMSG postulates and the neutral geometry theorems provided in the previous section. In the SMSG axiom list,
More informationSHAPE, SPACE AND MEASURES
SHAPE, SPACE AND MEASURES Pupils should be taught to: Use accurately the vocabulary, notation and labelling conventions for lines, angles and shapes; distinguish between conventions, facts, definitions
More informationChapter 1: Essentials of Geometry
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
More informationGeometry Review. Here are some formulas and concepts that you will need to review before working on the practice exam.
Geometry Review Here are some formulas and concepts that you will need to review before working on the practice eam. Triangles o Perimeter or the distance around the triangle is found by adding all of
More information11 th Annual HarvardMIT Mathematics Tournament
11 th nnual HarvardMIT Mathematics Tournament Saturday February 008 Individual Round: Geometry Test 1. [] How many different values can take, where,, are distinct vertices of a cube? nswer: 5. In a unit
More informationLet s Talk About Symmedians!
Let s Talk bout Symmedians! Sammy Luo and osmin Pohoata bstract We will introduce symmedians from scratch and prove an entire collection of interconnected results that characterize them. Symmedians represent
More informationGeometry Chapter 1 Vocabulary. coordinate  The real number that corresponds to a point on a line.
Chapter 1 Vocabulary coordinate  The real number that corresponds to a point on a line. point  Has no dimension. It is usually represented by a small dot. bisect  To divide into two congruent parts.
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationPERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various twodimensional figures.
PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various twodimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the
More informationLesson 18: Looking More Carefully at Parallel Lines
Student Outcomes Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180. They learn how to prove the alternate interior angles theorem using
More informationQuadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid
Quadrilaterals Properties of a parallelogram, a rectangle, a rhombus, a square, and a trapezoid Grade level: 10 Prerequisite knowledge: Students have studied triangle congruences, perpendicular lines,
More informationGeometry  Semester 2. Mrs. DayBlattner 1/20/2016
Geometry  Semester 2 Mrs. DayBlattner 1/20/2016 Agenda 1/20/2016 1) 20 Question Quiz  20 minutes 2) Jan 15 homework  selfcorrections 3) Spot check sheet Thales Theorem  add to your response 4) Finding
More informationIntro to Circles Formulas Area: Circumference: Circle:
Intro to ircles Formulas rea: ircumference: ircle: Key oncepts ll radii are congruent If radii or diameter of 2 circles are congruent, then circles are congruent. Points with respect to ircle Interior
More informationSan Jose Math Circle February 14, 2009 CIRCLES AND FUNKY AREAS  PART II. Warmup Exercises
San Jose Math Circle February 14, 2009 CIRCLES AND FUNKY AREAS  PART II Warmup Exercises 1. In the diagram below, ABC is equilateral with side length 6. Arcs are drawn centered at the vertices connecting
More information