# Section 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages : 1-18

Size: px
Start display at page:

Download "Section 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18"

Transcription

1 Chapter 9 Circles

2 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, theorems and corollaries in this chapter. D. Recognize circumscribed and inscribed polygons. E. Prove statements involving circumscribed and inscribed polygons. F. Solve problems involving circumscribed and inscribed polygons. G. Understand and apply theorems related to tangents, radii, arcs, chords, and central angles.

3 Section 9-1 Basic Terms: Tangents, Arcs and Chords Homework Pages : 1-18

4 Objectives A. Understand and apply the terms circle, center, radius, chord, secant, diameter, tangent, point of tangency, and sphere. B. Understand and apply the terms congruent circles, congruent spheres, concentric circles, and concentric spheres. C. Understand and apply the terms inscribed in a circle and circumscribed about a polygon. D. Correctly draw inscribed and circumscribed figures.

5 Circular Logic On a piece of paper, accurately draw a circle. What method did you use to make sure you drew a circle? Circle set of all coplanar points that are a given distance (radius) from a given point (center). Basic Parts: Radius Distance from the center of a circle to any single point on the circle. Center Point that is equidistant from all points on the circle. Indicated by symbol P Circle with center P Contrast a circle to a sphere: Sphere set of all points in space a given distance (radius) from a given point (center)

7 Lines and Line Segments Related to Circles Chord segment whose endpoints lie on a circle. Diameter a chord that passes through the center. Secant line that contains a chord. Tangent line in the plane of a circle that intersects the circle at exactly one point. Point of Tangency the point of intersection between a circle and a tangent to the circle.

8 Segments & Lines Chord Diameter Secant Tangent Point of Tangency

9 Circular Relationships Concentric circles coplanar circles with the same center Concentric Spheres spheres with the same center Congruent Circles circles with congruent radii Congruent Spheres spheres with congruent radii

10 Concentric Circles

11 Congruent Circles

12 A Figure Within a Figure Circumscribed About A Polygon Inscribed In a Circle all of the vertices of the polygon lie on a circle

14 Inscribed In A Circle

15 Sample Problems 1. Draw a circle and several parallel chords. What do you think is true of the midpoints of all such chords?

16 Sample Problems 3. Draw a right triangle inscribed in a circle. What do you know about the midpoint of the hypotenuse? Where is the center of the circle? If the legs of the right triangle are 6 and 8, find the radius of the circle. 8 6

17 Sample Problems 5. The radii of two concentric circles are 15 and 7. A diameter AB of the larger circle intersects the smaller circle at C and D. Find two possible values for AC.

18 Sample Problems 7. Draw a circle with an inscribed trapezoid.

19 Sample Problems Draw a circle and inscribe the polygon named. 9. a parallelogram 11. a quadrilateral PQRS with PR a diameter

20 Sample Problems For each draw a O with radius 12. Then draw OA and OB to form an angle with the measurement given. Find AB. 13. m AOB = m AOB = Q and R are congruent circles that intersect at C and D. CD is the common chord of the circles. What kind of quadrilateral is QDRC? Why? CD must be the perpendicular bisector of QR. Why? If QC = 17 and QR = 30, find CD. C Q R D

21 Section 9-2 Tangents Homework Pages : 1-18 Excluding 14

22 Objectives A. Understand and apply the terms external common tangent and internal common tangent. B. Understand and apply the terms externally tangent circles and internally tangent circles. C. Understand and apply theorems and corollaries dealing with the tangents of circles.

23 External Common Tangent External Common Tangent a line that is tangent to two coplanar circles and doesn t intersect the segment joining the centers of the circles.

24 Internal Common Tangent Internal Common Tangent a line that is tangent to two coplanar circles and intersects the segment joining the centers of the circles.

25 Tangent Circles Externally Tangent Circles coplanar circles that are tangent to the same line at the same point and the centers are on opposite sides of the line. Internally Tangent Circles coplanar circles that are tangent to the same line at the same point and the centers are on the same side of the line.

26 Externally Tangent Circles Externally Tangent Circles Center Center

27 Internally Tangent Circles Internally Tangent Center Center Point of Tangency

28 Theorem 9-1 If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency

29 Theorem 9-1 Corollary 1 Tangents to a circle from a point are congruent.

30 tangent line Theorem 9-2 If a line in a plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle.

31 Sample Problems JT is tangent to O at T. 1. If OT = 6 and JO = 10, JT =? 3. If m TOJ = 60 and OT = 6, JO =? J K T O 5. The diagram shows tangent lines and circles. Find PD. A 8.2 P D B C

32 Sample Problems 7. What do you think is true of the common external tangents AB and CD? Prove it. Will the results to this question still be true if the circles are congruent? B A Z D C

33 Sample Problems 9. Draw O with perpendicular radii OX and OY. Draw tangents to the circle at X and Y. If the tangents meet at Z, what kind of figure is OXZY? Explain. If OX = 5, find OZ. 11. Given: RS is a common internal tangent to A and B. Explain why AC RC BC SC R A C S B

34 Sample Problems 13. State the theorem which would describe the relationship between the planes tangent to a sphere at either end of a diameter. 15. PA, PB, and RS are tangents. Explain why PR + RS + SP = PA + PB A R C P B S

35 Sample Problems 17. JK is tangent to P and Q. JK =? J K P Q

36 Sample Problems 19. Given two tangent circles; EF is a common external tangent. Prove something about G. Prove something about EHF. E G F H

37 Section 9-3 Arcs and Central Angles Homework Pages : 1-20 Excluding 12

38 Objectives A. Understand and apply the term central angle. B. Understand and apply the terms major arc, minor arc, adjacent arcs, congruent arcs, and intercept arc. C. Understand and utilize the Arc Addition Postulate. D. Understand and apply the theorem of congruent minor arcs.

39 Central Angle an angle whose vertex lies on the center of a circle. Central Angle

40 Not Noah s Arc Arc an unbroken part of a circle. Types of arcs: Major arc Minor arc Adjacent arcs Congruent arcs Intercepted arc (covered in section 9-5) The symbol for the measurement of an arc is: mab measurement of arc AB

41 Adjacent Arcs arcs of the same circle that have exactly one point in common.

42 Congruent Arcs arcs in the same circle or congruent circles that have the same measurement. Congruent Arcs Same Length

43 Intercepted Arc the arc between the sides of an inscribed angle Intercept Arc Inscribed Angle

44 Minor Arc an unbroken part of a circle that measures less the 180.

45 Measure of a minor arc = measure of its central angle. 79 Measure of a minor arc = 79

46 Major Arc an unbroken part of a circle that measures more than 180 and less than 360.

47 Measure of major arc = measure of the minor arc

48 Semicircle an unbroken part of a circle that measures exactly 180 degrees. Semicircle arcs whose endpoints are the endpoints of a diameter.

49 Postulate 16 Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs. B A m A B m B C m A C C

50 Theorem 9-3 In the same circle or congruent circles, two minor arcs are congruent if and only if their central angles are congruent.

51 Sample Problems Find the measure of the central

52 Sample Problems 7. At 11 o clock the hands of a clock form an angle of? 9. Draw a circle. Place points A, B and C on it in such positions that ma B mb C ma C

53 Sample Problems C OC, OB, and OA are all radii. So OC = OB = OA If m COB 42, then m COA 138. A O B Since OC OA, then AOC is isosceles. Since AOC is isosceles, m ACO m CAO. D m AOC m CAO m ACO 138 (2 m CAO) 180 (2 m CAO) 42 m CAO

54 Sample Problems mc B mb D p 30 28?? q m COD?? 100?? A O C B m CAD??? 52? D

55 Sample Problems 15. Given: WZ is a diameter of O; mw X mx Y Prove: m Z = n n O W X Z Y The latitude of a city is given. Find the radius of this circle of latitude. 17. Milwaukee, Wisconsin; 43 N 19. Sydney, Australia; 34 S

56 Section 9-4 Arcs and Chords Homework Pages : 1-22

57 Objectives A. Understand the term arc of a chord. B. Understand and apply theorems relating arcs and chords to circles. C. Use the theorems related to arcs and chords to solve problems involving circles.

58 Arc of a Chord Arc of a Chord the minor arc created by the endpoints of the chord. Arc of a Chord Chord

59 Theorem 9-4 In the same circle or congruent circles: (1) Congruent arcs have congruent chords. (2) Congruent chords have congruent arcs.

60 Theorem 9-5 A diameter that is perpendicular to a chord bisects the chord and its arc.

61 Theorem 9-6 In the same circle or congruent circles: (1) Chords equally distant from the center (or centers) are congruent. (2) Congruent chords are equally distant from the center (or centers). A E C D B AB = CD = EF F

62 X Y M 3 5 O 1. XY =? Sample Problems OMY is a Therefore, pattern right triangle. MY 4. A diameter that is perpendicular to a chord bisects the chord and the arc. XY = 2MY = 2(4) = 8

63 Sample Problems O R S T 3. OT = 9, RS = 18 OR =?

64 Sample Problems A B A B O C O D 5. mb C? C D 80

65 Sample Problems A B O 7. m AOB = 60; AB = 24 OA =? A M B O C D N 9. AB = 18; OM = 12 ON = 10; CD =?

66 Sample Problems 11. Sketch a circle O with radius 10 and chord XY, 8. How far is the chord from O? 13. Sketch a circle P with radius 5 and chord AB that is 2 cm from P. Find the length of AB. 15. Given: J K Prove: J Z K Z Z J K

67 Sample Problems K 17. OJ = 10, JK =? O 120 J 19. A plane 5 cm from the center of a sphere intersects the sphere in a circle with diameter 24 cm. Find the diameter of the sphere. 21. Use trigonometry to find the measure of the arc cut off by a chord 12 cm long in a circle of radius 10 cm.

68 Section 9-5 Inscribed Angles Homework Pages : 1-24 (no 14)

69 Objectives A. Understand and apply the terms inscribed angle and intercepted arc. B. Understand and apply the theorems and corollaries associated with inscribed angles and intercepted arcs of circles. C. Use the theorems and corollaries associated with inscribed angles and intercepted arcs to solve problems involving circles.

70 Inscribed Angle Inscribed Angle an angle whose vertex lies on a circle and whose sides contain chords of the circle. Chords Inscribed Angle Vertex

71 Intercepted Arc Intercepted Arc an arc formed on the interior of an angle. Intercepted Arc Inscribed Angle

72 Theorem 9-7 The measure of an inscribed angle is equal to half the measure of its intercepted arc. ½(x ) x

73 Theorem 9-7 Corollary 1 If two inscribed angles intercept the same arc, then the angles are congruent.

74 Theorem 9-7 Corollary 2 An angle inscribed in a semicircle is a right angle.

75 Theorem 9-7 Corollary 3 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. B m A + m C = 180 m B + m D = 180 A C D

76 Theorem 9-8 The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc. A m ABC = 1 mab 2 B C

77 z y O x 50 Sample Problems What else do you know? Is there a diameter? What is the measure of a semicircle? x + 50 = 180 Why? x = 30 What else do you know? The measure of an inscribed angle is equal to half the measure of its intercepted arc. y = ( ½ ) 50 y = 25 z = ( ½ ) 30 z = x y z What else do you know? If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary x = 180 x = y = 180 y = 100 What else do you know? z = 2(x ) Why? z = 2(110 ) z = 100

78 Sample Problems y z 50 x z What else do you know? Why? z + z + 50 = 180 Why? z = 65 Why? What else do you know? Why? The measure of an inscribed angle is equal to half the measure of its intercepted arc. What else do you know? The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc. x = ( ½ ) 100 x = 50 y =? z = ( ½ ) y 65 = ( ½ ) y y = 130

79 7. x y 76 z Sample Problems y What else do you know? If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. x + 76 = 180 x = 104 What else do you know? In the same or congruent circles, congruent chords have congruent arcs. What else do you know? 2x = y + y Why? y = 104 Why? What else do you know? The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc. z = ( ½ ) y z = ( ½ ) 104 z = 52

80 Sample Problems 9. y z x x 90 What else do you know? Why? What else do you know? The measure of an inscribed angle is equal to half the measure of its intercepted arc. x = ( ½ ) 100 x = 50 y =? ( ½ ) y = x ( ½ ) y = 50 y = 100 z =? z = ( ½ ) (360 - ( )) Why? z = 35

81 Sample Problems 17. Draw an inscribed quadrilateral ABCD and its diagonals intersecting at E. Name two pairs of similar triangles.

82 Sample Problems ABCD is an inscribed quadrilateral. 19. m A = x, m B = 2x, m C = x Find x and m D. 21. m D = 75, ma B Find x and m A. x 2, mb C 5x and mc D 6x. 23. Equilateral ABC is inscribed in a circle. P and Q are midpoints of arcs BC and CA respectively. What kind of figure is quadrilateral AQPB? Why?

83 Section 9-6 Other Angles Homework Pages : 1-24

84 Objectives A. Understand and apply the theorem relating to two chords intersecting inside of a circle. B. Understand and apply the theorem relating two secants, two tangents or a secant and a tangent of a circle. C. Use these theorems to solve problems relating to circles.

85 Theorem 9-9 The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. A x C D x mab m C D 2 B

86 Theorem 9-10 The measure of the angle formed by two secants, two tangents or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs. A A A B x C x D mab m C D 2 B x x C B mab m B C 2 x x C ma B C mac 2

87 Hints to help you remember these theorems! If the vertex of the angle in question is INSIDE of the circle, ADD the intercepted arcs and divide by two to get the measure of the angle. Can a measure of an angle EVER be negative? If the vertex of the angle in question is OUTSIDE of the circle, SUBTRACT the smaller intercepted arc from the larger intercepted arc and divide the result by two to get the measure of the angle. R 1 ADD! U SUBTRACT! B A T S C T mrt m 1 mus 2 mct mbt m A 2

88 BZ AC m BC mcd m DE Sample Problems 1-10: Find the measure of each angle. What should you do first? is a tangent line. is a diameter A What type of angle is angle 1? What type of angle is angle 3? 1 m Where is the vertex of angle 5? B 9 10 m 9? m AB? m 1? m 3? O 4 m 5? 5 6 D 1 m m Where is the vertex of angle 7? 2 E Z m 1 90 m 3 25 m 5 55 m 9 90 m 7 35 C 30 7 m 7?

89 Complete. 11. If 13. m RT then m 1 =? Sample Problems 80 and mus 40, If m 1 50 and m RT 70, then mus? # 11 What should you do first? Where is the vertex of the angle? The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. m RT mus m R T 1 U # 13 What should you do first? Where is the vertex of the angle? The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. m RT mus 70 mus m mus mus 30 S

90 Sample Problems Segment AT is a tangent line. 15. If mct 110 and m BT 50, then m A =? C B 50 T A # 15 What should you do first? What else do you know? What type of lines contain segments AC and AT? Where is vertex of the angle? The measure of the angle formed by two secants, two tangents or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs. 110 mct m BT m A

91 Sample Problems Segment AT is a tangent line. 17. If m A 35 and mct 110, then mbt? C B 35 T A # 17 What should you do first? What else do you know? What type of lines contain segments AC and AT? Where is the vertex of the angle? The measure of the angle formed by two secants, two tangents or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs. m A 110 mct m BT m BT m BT m BT 40

92 Sample Problems PX and PY are tangent segments. X 19. If mxy 90, then m P? P 90 Z # 19 What should you do first? What else do you know? mxzy? mxzy 360 mxy Where is the vertex of the angle? What else do you know? Y The measure of the angle formed by two secants, two tangents or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs. m XZY m XY m P

93 Sample Problems PX and PY are tangent segments. X 21. If m P 65, then mxy? P 65 Z # 21 What should you do first? What else do you know? mxzy? mxzy 360 mxy Y The measure of the angle formed by two secants, two tangents or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs. Where is the vertex of the angle? m XZY m XY What else do you know? m P mxy m XY mXY 2mXY 230 mxy

94 Sample Problems 23. A quadrilateral circumscribed about a circle has angles 80, 90, 94 and 96. Find the measures of the four nonoverlapping arcs determined by the points of tangency. 27. Write an equation involving a, b and c. a c b

95 Section 9-7 Circles and Lengths of Segments Homework Pages : 1-26

96 Objectives A. Understand and apply theorems relating the product of segments of chords, secants, and tangents of a circle. B. Use these theorems to solve problems involving circles.

97 Theorem 9-11 When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. a y x b (a) (b) = (x)(y)

98 Theorem 9-12 When two secants are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment. a b y x (a) (b) = (x)(y)

99 Theorem 9-13 When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment equals the square of the tangent segment a b (a)(b) = x 2 x

100 Sample Problems Solve for x x # 1 What else do you know? When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. 4(x) = 5(8) x = # 3 What else do you know? When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment equals the square of the tangent segment. 3 x 2 x x

101 x Sample Problems Solve for x. 4 # 5 What else do you know? When two secants are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment. (8)(5) = (4 + x)(4) 40 = x 24 = 4x x = x 5 # 7 What else do you know? When two secants are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment. (x)(5) = (10)(4) x = 8 4

102 Sample Problems Solve for x x x # 9 What else do you know? When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment equals the square of the tangent segment x x x 100 4x x x 2 25 x x 25 x 5 x 5

103 Sample Problems Chords AB and CD intersect at P. 13. AP = 6, BP = 8, CD = 16 DP =? # 13 What should you do first? What else do you know? When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. (6)(8) = (DP)(16 DP) DP 2 b b 4ac 2a DP DP DP 2 C A P Now what? 6 DP D B 16DP ( 16) ( 16) 4(1)(48) DP DP 2(1) DP DP or DP 12 or 4 2 2

104 Sample Problems Chords AB and CD intersect at P. 15. AB = 12, CP = 9, DP = 4 BP =? # 15 What should you do first? What else do you know? When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. (4)(9) = (BP)(12 BP) BP 2 b b 4ac 2a BP BP BP 2 C BP A 2 9 P 12 2 ( 12) ( 12) 4(1)(36) BP 2(1) 12 BP D B 12BP 36 0

105 Sample Problems Segment PT is tangent to the circle. 17. PT = 6, PB = 3 AB =? # 17 What should you do first? What else do you know? A T B 6 3 D P When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment equals the square of the tangent segment AP 36 3AP AP 12 C AP AB BP 12 AB 3 AB 9

106 Sample Problems Segment PT is tangent to the circle. 19. PD = 5, CD = 7, AB = 11, PB =? # 19 What should you do first? What else do you know? A 11 T 7 B D 5 P When two secants are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment. BP AP DP CP BP 2 BP 11 BP BP 60 0 BP 15 BP 4 0 Can BP = -15? Can BP = 4? BP = 4 C 2 11BP BP 60 BP 15 or 4

107 Sample Problems 23. A bridge over a river has the shape of a circular arc. The span of the bridge is 24 meters. The midpoint of the arc is 4 meters higher than the endpoints. What is the radius of the circle that contains the arc.

108 Chapter 9 Circles Review Homework Page 371: 2-16 evens

### Chapter 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

### Circle 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

### Chapters 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

### Name Date Class. Lines and Segments That Intersect Circles. AB and CD are chords. Tangent Circles. Theorem Hypothesis Conclusion

Section. Lines That Intersect Circles Lines and Segments That Intersect Circles A chord is a segment whose endpoints lie on a circle. A secant is a line that intersects a circle at two points. A tangent

### Circle 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,

### Definitions, Postulates and Theorems

Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven

### Conjectures. Chapter 2. Chapter 3

Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

### DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent

### Geometry 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.

### Unit 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,

### Geometry 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.

### For 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

### Geometry 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,

### Math 531, Exam 1 Information.

Math 531, Exam 1 Information. 9/21/11, LC 310, 9:05-9:55. Exam 1 will be based on: Sections 1A - 1F. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/531fa11/531.html)

### 2006 Geometry Form A Page 1

2006 Geometry Form Page 1 1. he hypotenuse of a right triangle is 12" long, and one of the acute angles measures 30 degrees. he length of the shorter leg must be: () 4 3 inches () 6 3 inches () 5 inches

### CCGPS 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 M9-1.G..1 Prove that all circles are similar. M9-1.G.. Identify and describe relationships

### 39 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

### Contents. 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..................................

### Conjectures for Geometry for Math 70 By I. L. Tse

Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:

### GEOMETRY CONCEPT MAP. Suggested Sequence:

CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons

### Geometry Regents Review

Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 18, 2010 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of

### Unit 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. (10-1) Circles and Circumference

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

### Chapter 11. Areas of Plane Figures You MUST draw diagrams and show formulas for every applicable homework problem!

Chapter 11 Areas of Plane Figures You MUST draw diagrams and show formulas for every applicable homework problem! Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret

### Tangent 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

### Class-10 th (X) Mathematics Chapter: Tangents to Circles

Class-10 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

### Lesson 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

### Area. Area Overview. Define: Area:

Define: Area: Area Overview Kite: Parallelogram: Rectangle: Rhombus: Square: Trapezoid: Postulates/Theorems: Every closed region has an area. If closed figures are congruent, then their areas are equal.

### San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS

San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS Recall that the bisector of an angle is the ray that divides the angle into two congruent angles. The most important results about angle bisectors

### Postulate 17 The area of a square is the square of the length of a. Postulate 18 If two figures are congruent, then they have the same.

Chapter 11: Areas of Plane Figures (page 422) 11-1: Areas of Rectangles (page 423) Rectangle Rectangular Region Area is measured in units. Postulate 17 The area of a square is the square of the length

### Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On

### Angles 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,

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### 56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points.

6.1.1 Review: Semester Review Study Sheet Geometry Core Sem 2 (S2495808) Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which

### Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.

Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.

### 1. 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

### Algebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids

Algebra III Lesson 33 Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Quadrilaterals What is a quadrilateral? Quad means? 4 Lateral means?

### CHAPTER 8 QUADRILATERALS. 8.1 Introduction

CHAPTER 8 QUADRILATERALS 8.1 Introduction You have studied many properties of a triangle in Chapters 6 and 7 and you know that on joining three non-collinear points in pairs, the figure so obtained is

### Warm-up 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 Warm-up 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

### 5.1 Midsegment Theorem and Coordinate Proof

5.1 Midsegment Theorem and Coordinate Proof Obj.: Use properties of midsegments and write coordinate proofs. Key Vocabulary Midsegment of a triangle - A midsegment of a triangle is a segment that connects

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 20, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

### QUADRILATERALS CHAPTER 8. (A) Main Concepts and Results

CHAPTER 8 QUADRILATERALS (A) Main Concepts and Results Sides, Angles and diagonals of a quadrilateral; Different types of quadrilaterals: Trapezium, parallelogram, rectangle, rhombus and square. Sum of

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2009 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

### CSU Fresno Problem Solving Session. Geometry, 17 March 2012

CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news

### Geometry Enduring Understandings Students will understand 1. that all circles are similar.

High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,

### Test 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

### 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.

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

### Chapter Review. 11-1 Lines that Intersect Circles. 11-2 Arcs and Chords. Identify each line or segment that intersects each circle.

HPTR 11-1 hapter Review 11-1 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

### 1 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,

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, June 17, 2010 1:15 to 4:15 p.m., only Student Name: School Name: Print your name and the name of your

### New York State Student Learning Objective: Regents Geometry

New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students

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

### Geometry Module 4 Unit 2 Practice Exam

Name: Class: Date: ID: A Geometry Module 4 Unit 2 Practice Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which diagram shows the most useful positioning

### 3.1 Triangles, Congruence Relations, SAS Hypothesis

Chapter 3 Foundations of Geometry 2 3.1 Triangles, Congruence Relations, SAS Hypothesis Definition 3.1 A triangle is the union of three segments ( called its side), whose end points (called its vertices)

### IMO Geomety Problems. (IMO 1999/1) Determine all finite sets S of at least three points in the plane which satisfy the following condition:

IMO Geomety Problems (IMO 1999/1) Determine all finite sets S of at least three points in the plane which satisfy the following condition: for any two distinct points A and B in S, the perpendicular bisector

Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are

### Geo 9 1 Circles 9-1 Basic Terms associated with Circles and Spheres. Radius. Chord. Secant. Diameter. Tangent. Point of Tangency.

Geo 9 1 ircles 9-1 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

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### Geometry. Higher Mathematics Courses 69. Geometry

The fundamental purpose of the course is to formalize and extend students geometric experiences from the middle grades. This course includes standards from the conceptual categories of and Statistics and

### 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

### Lesson 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

### How To Understand The Theory Of Ircles

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

### POTENTIAL REASONS: Definition of Congruence:

Sec 6 CC Geometry Triangle Pros Name: POTENTIAL REASONS: Definition Congruence: Having the exact same size and shape and there by having the exact same measures. Definition Midpoint: The point that divides

### 1. 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

### Intro 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

### alternate interior angles

alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

### CIRCLE 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

### Conjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)

Mathematical Sentence - a sentence that states a fact or complete idea Open sentence contains a variable Closed sentence can be judged either true or false Truth value true/false Negation not (~) * Statement

### GEOMETRY. 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

### Georgia Online Formative Assessment Resource (GOFAR) AG geometry domain

AG geometry domain Name: Date: Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (1 of 36 ) 1. Amy drew a circle graph to represent

### Unit 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

### Chapter 7 Quiz. (1.) Which type of unit can be used to measure the area of a region centimeter, square centimeter, or cubic centimeter?

Chapter Quiz Section.1 Area and Initial Postulates (1.) Which type of unit can be used to measure the area of a region centimeter, square centimeter, or cubic centimeter? (.) TRUE or FALSE: If two plane

### GEOMETRY COMMON CORE STANDARDS

1st Nine Weeks Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,

### Geometry - Semester 2. Mrs. Day-Blattner 1/20/2016

Geometry - Semester 2 Mrs. Day-Blattner 1/20/2016 Agenda 1/20/2016 1) 20 Question Quiz - 20 minutes 2) Jan 15 homework - self-corrections 3) Spot check sheet Thales Theorem - add to your response 4) Finding

### Geometry Course Summary Department: Math. Semester 1

Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give

### Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Class: Date: ID: A Q3 Geometry Review Multiple Choice Identify the choice that best completes the statement or answers the question. Graph the image of each figure under a translation by the given

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

### Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)

### The 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

### Grade 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

### Lesson 1.1 Building Blocks of Geometry

Lesson 1.1 Building Blocks of Geometry For Exercises 1 7, complete each statement. S 3 cm. 1. The midpoint of Q is. N S Q 2. NQ. 3. nother name for NS is. 4. S is the of SQ. 5. is the midpoint of. 6. NS.

### CHAPTER 1. LINES AND PLANES IN SPACE

CHAPTER 1. LINES AND PLANES IN SPACE 1. Angles and distances between skew lines 1.1. Given cube ABCDA 1 B 1 C 1 D 1 with side a. Find the angle and the distance between lines A 1 B and AC 1. 1.2. Given

### For each Circle C, find the value of x. Assume that segments that appear to be tangent are tangent. 1. x = 2. x =

Name: ate: Period: Homework - Tangents For each ircle, find the value of. ssume that segments that appear to be tangent are tangent. 1. =. = ( 5) 1 30 0 0 3. =. = (Leave as simplified radical!) 3 8 In

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXMINTION GEOMETRY Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

### CK-12 Geometry: Parts of Circles and Tangent Lines

CK-12 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.

### Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment

Geometry Chapter 1 Section Term 1.1 Point (pt) Definition A location. It is drawn as a dot, and named with a capital letter. It has no shape or size. undefined term 1.1 Line A line is made up of points

### Shape, Space and Measure

Name: Shape, Space and Measure Prep for Paper 2 Including Pythagoras Trigonometry: SOHCAHTOA Sine Rule Cosine Rule Area using 1-2 ab sin C Transforming Trig Graphs 3D Pythag-Trig Plans and Elevations Area

### Final Review Geometry A Fall Semester

Final Review Geometry Fall Semester Multiple Response Identify one or more choices that best complete the statement or answer the question. 1. Which graph shows a triangle and its reflection image over

### GEOMETRY B: CIRCLE TEST PRACTICE

Class: Date: GEOMETRY B: CIRCLE TEST PRACTICE Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the measures of the indicated angles. Which statement

### Chapter 8 Geometry We will discuss following concepts in this chapter.

Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles

### Equation 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

### IMO 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.

### PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the

### GEOMETRY OF THE CIRCLE

HTR GMTRY F TH IRL arly geometers in many parts of the world knew that, for all circles, the ratio of the circumference of a circle to its diameter was a constant. Today, we write d 5p, but early geometers

### MATH 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

### GEOMETRY. Chapter 1: Foundations for Geometry. Name: Teacher: Pd:

GEOMETRY Chapter 1: Foundations for Geometry Name: Teacher: Pd: Table of Contents Lesson 1.1: SWBAT: Identify, name, and draw points, lines, segments, rays, and planes. Pgs: 1-4 Lesson 1.2: SWBAT: Use