# Geometry Unit 10 Notes Circles. Syllabus Objective: The student will differentiate among the terms relating to a circle.

Save this PDF as:

Size: px
Start display at page:

Download "Geometry Unit 10 Notes Circles. Syllabus Objective: 10.1 - The student will differentiate among the terms relating to a circle."

## Transcription

1 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, called the center. adius the distance from the center to a point on the circle. iameter the distance across a circle, through the center. The diameter is twice the radius. hord a segment whose endpoints are points on the circle. Secant a line that intersects a circle in two points. Tangent a line in the plane of a circle that intersects the circle in exactly one point. oncentric circles coplanar circles that have a common center. Some real life examples are targets and Target! oint of tangency the point at which the tangent line intersects the circle. Syllabus Objective: The student will explore relationships among circles and external lines or rays. Example: Find the length of chord. O 6 0 N x O = O = 6. raw ON. ON bisects ; ON bisects O. Using the properties of triangles, ON = 3 and N = 3 3, so ( ) = 3 3 = 6 3. Unit 0 ircles age of 0

2 Theorem: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Theorem: In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. Example: and are tangent to O at and, respectively. O If = and = 8, then O =?. Since passes through the center, it is perpendicular to the tangent line. The diameter is, so the radius 8 6, ( O ) the ythagorean Theorem to find O. + =O O = 0. is 6. Use Theorem: If two segments from the same exterior point are tangent to a circle, then they are congruent. {Think of an ice cream cone, the two sides should be equal where they meet the ice cream.} Example: a) Find the perimeter of the polygon. I H.5 G J F E 3.5 = J = 3, = =.5, E = EF =, GF = GH =, IH = IJ =.5. dd them all together, = 0 units. Syllabus Objective: The student will solve problems involving arcs, chords, and radii of a circle. entral angle an angle whose vertex is the center of a circle. Minor arc part of a circle that measures less than 80. Major arc part of a circle that measures between 80 and 360. Unit 0 ircles age of 0

3 Semicircle an arc whose endpoints are the endpoints of a diameter of the circle. semicircle measures exactly 80. rc ddition ostulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Theorem: In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Examples: a) In E, find the measure of the angle or the arc named. 80 E Solutions: m = m E = 80. m = m = 70. m = m + m = = 50. m = 360 m = = 90. b) In with diameter S, find the measure of the angle or the arc named. S T 45 Q 60 Solutions: Solutions: Q 60 SQ 40 ST 45 T 35 SQ 80 T 35 SQ 0 ST 35 SQ 0 TSQ 65 S 80 Theorem: If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. Theorem: If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. Unit 0 ircles age 3 of 0

4 Examples: a) In, SQ = and T = 8. Find. S T Q ST = TQ = SQ = ( ) = 6, ( S) = ( ST ) + ( T ) 6 8, ΔTS is a right triangle, so S = 0. = +, ( S ) is a diameter, so = = 0. b) mgmj = 00, mgk =?. G K H Q M J Since KM is perpendicular to GJ, it also bisects GJ and GJ = 60, mgj = 60, mgk = mgj = ( 60 ) = 80. c) and are tangent to O at and, respectively., if m = 50, then m O =? Δ must measure = 30 to be shared evenly by O and. m = 65, m O = = 5. m O = 5. Theorem: In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Unit 0 ircles age 4 of 0

5 Example: In O, FL = 3, GO = 5, O = 4. Find HJ. G L O F J H L is the midpoint of FG, so LG = 3. ( OL) ( OL) In rt. Δ OLG, 5 = 3 +. Then 5 9 = 6 =, OL = 4. Since O = 4, OL = O an d FG HJ, HJ = FG = FL = 6. Syllabus Objective: 0. - The student will solve problems involving angles, arcs, or sectors of circles. One easy way for students to remember the angle measures relationships: have them imagine a rubber band being held down on two points of the circle. s the rubber band is stretched to the center of the circle it will form an angle, which is equal to the arc measure. Stretched some more and it will be thinner (one-half the sum of the arcs). Stretched even further, to the edge of the circle and it will be thinner (one-half the arc). nd stretched further again, even thinner still (one-half the difference of the arcs). Inscribed angle an angle whose vertex is on a circle and whose sides contain chords of the circle. Intercepted arc the arc that lies in the interior of an inscribed angle and has endpoints on the angle. Inscribed polygon drawn inside. polygon is inscribed in a circle if all its vertices lie on the circle. ircumscribed drawn around. In the above definition, the circle is circumscribed about the polygon. Measure of an Inscribed ngle Theorem: If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. { angle = arc } Unit 0 ircles age 5 of 0

6 Example: Find the values of x, y, and z. 90 x y x = ( 80 ) = 40, y = 55, y = 0, z = = 80. ( 0) z Theorem: If two inscribed angles of a circle intercept the same arc, then the angles are congruent. Example: If m = 0, then m =? and m =?. Since and both intercept. m = m = m = 60. Theorem: If a right triangle is inscribed is inscribed in a circle, then the hypotenuse is a diameter of the circle. onversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. { angle = arc } Example: If E is a diameter, then E is a? and m 3 =?. 3 E E is a semicir cle, so m 3 = 90. Unit 0 ircles age 6 of 0

7 Theorem: quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. Example: If m 4 = 75, then m 5 =?. 4 Since opposite angles of an inscribed quadrilateral are supplementary, m 5 = = Syllabus Objective: The student will solve problems involving secant segments and tangent segments for a circle. T angent segment a segment that is tangent to a circle at an endpoint. tangent segment Secant segment a segment that intersects a circle in two points, with one point as an endpoint of the segment. secant segment E xternal segment the part of the secant segment that is not inside the circle. external segment Theorem: If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. { angle = arc } Unit 0 ircles age 7 of 0

8 Example: If X is tangent to and m XZY = 60, find mxy and m XY. Z X mxy = 360 mxzy = = 00. m XY = mxy = ( 00) = 50. Y Theorem: If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of th e measures of the arcs intercepted by the angle and its vertical angle. { angle = ( arc + arc )} Exa mples: a) If mq = 45 and ms = 75, find m. Q m = ( ms + mq ) = ( ) = 60. S b) If m = 55 and ms = 80, find mq. m = ( ms + mq) 55 = ( 80 + mq) 0 = 80+ m Q 30 = mq. Theorem: If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the m easure of the angle formed is one half the difference of the measure of the intercepted arcs. { angle = ( larger arc smaller arc )} Unit 0 ircles age 8 of 0

9 Examples: a) If m = 00 and me = 40, find m. m = ( m me ) E = ( 00 40) = 30 Secant-Secant b) c) If m W = 65 and mxz = 70, find mxvy. Y m W = ( mxvy mxz ) Z 65 = ( mxvy 70) W 30 = mx VY = mx VY. X V Secant-Tangent If mqs = 40, find mqs and m. S mqs + mqs = 360, 40 + mqs = 360, mqs = 0. m = ( mqs mqs ), Q = ( 40 0 ) = 60. Tangent-Tangent Theorem: If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the products of the lengths of the segments of the other chord. Unit 0 ircles age 9 of 0

10 Examples: a) If = 4, = 6, and = 8, find. i = i ( ) = ( ) =. b) If = 4, = 9, and = 5, find. Let = x ; then = 5 x. i = i x ( ) = x ( x) = 5x x 5x + 36 = 0 ( x )( x ) 3 = 0 x = or x = 3 = or = 3. Theorem: If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment. {(entire length)(outside length)=(entire length)(outside length)} For both of the external segment theorems, students must remember; (entire length)(part outside) = (entire length)(part outside). This works for the tangent segments as well because the segment length is also the external segment length (tangent squared). Unit 0 ircles age 0 of 0

11 Examples: F E G a) If F = 7, EF = 3, and GF = 6, then HF =?. F ief = HF igf, ( 7)( 3) = ( HF )( 6) HF = 8.5. b) If HG = 8, GF = 7, and F =, then EF =?. F ief = HF igf, ( )( EF ) = ( + )( ) 8 7 7, EF = 5., Theorem: If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment. {(entire length)(outside length)=(entire length)(outside length)} H Examples: Q S T a) If T = 5 and S = 5, find Q. {(entire length)(outside length)=(entire length)(outside length)} T is = Q iq T is = Q ( )( ) = ( Q ) 5 5, 5 5 = Q. or, b) If Q = 6 and ST = 9, find S. Let S = x ; then T = x + 9, T is = ( Q ) ( x + )( ) =, 9 9 6, x + 9x 36 = 0, ( x + )( x 3) = 0, x = or x = 3, so S = 3. Unit 0 ircles age of 0

12 Syllabus Objective: The student will solve problems involving properties circles using algebraic techniques. Examples: a) Find the value of x. 4 3x+7 5x-9 4 Since the arcs are equal (congruent), the chords are congruent. 3x + 7 = 5x 9, 6 = x, x = 8. b) Find m and m. (x-8) (3x+) Since is inscribed in a circle, opposite angles are supplementary. m + m = 80, 3x + + x 8 = 80, 44x + 4 = 80, 44x = 76, x = 4. ( ) So, m = 34 + = 04 and ( ) m = 4 8 = 76. c) Find x and y. ssume that segments that appear to be tangent are tangent. ound to the nearest tenth if necessary. Two tangent segments are equal so, 5x-8 y 4 5x 8 = 7 3 x, 8x = 80, x = 0. adius is perpendicular to tangent line so, y + 39 = 4, y = 60 = x 39 Unit 0 ircles age of 0

13 d) Find x. J x+ x x+0 K x+8 M L JK ikl = K ikm ( )( ) ( )( x + 0 x = x + x = x x x x 0x = 9x + 8 x = 8. ) Syllabus Objective: The student will perform constructions involving special relationships within circles. {May require supplemental material} Example: onstruct a tangent to a given circle from a point on the circle. O t Given: oint on O. Step : raw O. Step : onstruct a perpendicular from a point ON the line, named t. Line t is tangent to O at. **Eggs** Give students the construction steps and see if they can recreate the shape. The beautifully smooth egg shown below is composed of circular segments that fit together continuously, so that there is no sudden change in the direction of the shell, where any two segments touch. The roblem : econstruct the egg yourself, using only a compass and a straightedge. Unit 0 ircles age 3 of 0

14 Notes : G F N E The order of construction would be:. raw two congruent circles, the second having its center on the circumference of the first.. Join their centers. ( ) 3. Join their points of intersection. (E the perpendicular bisector of ) 4. Mark the intersection of and E. (the mid-point of as N) 5. Mark off point on the line segment N, such that N = N = N. 6. Lengthen and to cut the inside of the original circles at F and G respectively. (and form Δ ) 7. Use compasses centered at, radius F, to draw minor circular FG. 8. Use compasses centered at N, radius N, to draw semi-circular. 9. Go over FG, G, and F to emphasize the required Egg shape. Syllabus Objective: The student will graph a circle and determine its equation. Standard equation of a circle a circle with radius r and center (h, k) has this standard equation: ( ) ( ) x h + y k = r Examples: Write the equation of the circle. a) enter at (, 8), radius 7. ( x h) ( y k) ( x ) y ( ) ( x ) ( y ) + = r, ( ) ( ) + 8 = 7, = 7. Unit 0 ircles age 4 of 0

15 b) enter at (, 4), passes through ( 6, 7 ) ( ). Use distance formula to find radius: center to point on circle. r = + ( 6 ) (7 4), r = + ( 4) (3), r = 6+ 9 = 5 = 5. ( x h) + ( y k) = r, ( x ( )) + ( y ) = ( ) ( x + ) + ( y 4) = , Unit 0 ircles age 5 of 0

16 This unit is designed to follow the Nevada State Standards for Geometry, S syllabus and benchmark calendar. It loosely correlates to hapter 0 of Mcougal Littell Geometry 004, sections The following questions were taken from the nd semester common assessment practice and operational exams for and would apply to this unit. Multiple hoice # ractice Exam (08-09) Operational Exam (08-09) 3. Which accurately describes a tangent?. segment whose endpoints are on the circle.. line that intersects a circle in two points and passes through the center of the circle.. segment having an endpoint on the circle and an endpoint at the center of the circle.. line that intersects a circle at exactly one point. 4. Use the figure below. Which accurately describes a secant?. line that intersects a circle at two points.. segment whose endpoints are on the circle.. segment having an endpoint on the circle and an endpoint at the center of the circle.. line that intersects a circle at exactly one point. Use the figure below. E G E G W ich of the following represent a secant? h. G. E.. 5. In circle S below, Q 3 S Which represents a chord?.... G Use circle O below. Q 86 O T T The m QT = 3, what is the measure of QT? Since m Q= 86, what is the measure of T? Unit 0 ircles age 6 of 0

17 6. In circle J below, L 56 Use circle J below. L J ( 3x 6) K J 66 ( x 4) What is the value of x? In K, =, mxy ( 7x 9) = 3( x + 5) mwz, and m XLY = 48. K What is the value of x? In, ms ( x 9) and m QS = 3. = +, mtu ( x 7) = +, W X S K L Q Z Y T U What is the value of x? In the figure below, m = 75 and m= 35, Q What is the value of x? In the figure below, mmn = 50, M J L m H = 3 and H K What is m? N What is the mjk? Unit 0 ircles age 7 of 0

18 9. Two tangents are drawn from point to circle H. N Two tangents are drawn from point to circle. H What conclusion is guaranteed by this diagram?. mn m N =. ΔHN is a right triangle.. HN is a rhombus.. HN is a kite. 0. ll of the segments shown in the figure below are tangents to N. T What conclusion is guaranteed by this diagram?. m m =. Δ is a right triangle. =. Δ is an isosceles triangle ll of the segments shown in the figure below are tangents to K. G 5 cm 6 cm N 0 cm K 8 cm W 4 cm V Given the measures in the figure above, what is the perimeter of quadrilateral?. 3 cm. 40 cm. 46 cm. 5 cm U 3 cm 3cm I E Given the measures in the figure above, what is the perimeter of quadrilateral GHI?. 8 cm. 6 cm. 3 cm. 36 cm H cm Unit 0 ircles age 8 of 0

19 . In K, NK = 3x + 4, KW = 5x 8, S = 5x 4, and KN KW. N In, FE, H = 4x +, and FE = 6x + 0. E G K F W H What is N? K is the diameter of O, and mjk ( 9( x ) 6) = +. O S =, mj ( 9x) What is the value of x? S is the diameter of M, m MT = ( x+ 5), m UM= ( 3x+ 5), and m SMT = 4x 0. S ( ) J K L What is the value of x? U M What is the value of x? T Unit 0 ircles age 9 of 0

20 48. In circle below, is tangent to at, and is tangent to at. What is the length of? In the figure below, is tangent to at and is tangent to at. ( x + 5) x 6 3( x 7) 6 5x 0 In the figure below, is tangent to at and is tangent to at. x Find the value of x In circle below, MN is tangent to N and M is tangent to at. M 3x N 7 at What is the value of x? In the figure below, is tangent to the circle at and S is a secant. S What is the value of x?. 48 cm. 84 cm. 4 3 cm. cm 8 cm V x 6 cm What is the length of M? In the figure below, is tangent to the circle at and is a secant. 6x What is the value of x?. cm. 5 cm. 5 cm. 5 cm x 0 cm 5 cm Unit 0 ircles age 0 of 0

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### Tangents to Circles. Circle The set of all points in a plane that are equidistant from a given point, called the center of the circle

10.1 Tangents to ircles Goals p Identify segments and lines related to circles. p Use properties of a tangent to a circle. VOULRY ircle The set of all points in a plane that are equidistant from a given

More information

### Warm 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 information

### circumscribed 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 information

### Geometry 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/chapter-10-circles-flashcardsflash-cards/).

More information

### 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

More information

### c.) If RN = 2, find RP. N

Geometry 10-1 ircles and ircumference. Parts of ircles 1. circle is the locus of all points equidistant from a given point called the center of the circle.. circle is usually named by its point. 3. The

More information

### 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.

More information

### Geometry 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 information

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

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 information

### 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

More information

### 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

More information

### 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

More information

### Circle geometry theorems

Circle geometry theorems http://topdrawer.aamt.edu.au/geometric-reasoning/big-ideas/circlegeometry/angle-and-chord-properties Theorem Suggested abbreviation Diagram 1. When two circles intersect, the line

More information

### 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

More information

### 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

More information

### GEOMETRY FINAL EXAM REVIEW

GEOMETRY FINL EXM REVIEW I. MTHING reflexive. a(b + c) = ab + ac transitive. If a = b & b = c, then a = c. symmetric. If lies between and, then + =. substitution. If a = b, then b = a. distributive E.

More information

### 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

More information

### 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

More information

### of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

2901 Clint Moore Road #319, Boca Raton, FL 33496 Office: (561) 459-2058 Mobile: (949) 510-8153 Email: HappyFunMathTutor@gmail.com www.happyfunmathtutor.com GEOMETRY THEORUMS AND POSTULATES GEOMETRY POSTULATES:

More information

### The 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 2013-2014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to

More information

### 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

More information

### 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

More information

### Centroid: 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 information

### 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

More information

### Geometry 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 information

### CIRCLE 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 information

### 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:

More information

### ABC is the triangle with vertices at points A, B and C

Euclidean Geometry Review This is a brief review of Plane Euclidean Geometry - symbols, definitions, and theorems. Part I: The following are symbols commonly used in geometry: AB is the segment from the

More information

### 10.5 and 10.6 Lesson Plan

Title: Secants, Tangents, and Angle Measures 10.5 and 10.6 Lesson Plan Course: Objectives: Reporting Categories: Related SOL: Vocabulary: Materials: Time Required: Geometry (Mainly 9 th and 10 th Grade)

More information

### Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

More information

### BC AB = AB. The first proportion is derived from similarity of the triangles BDA and ADC. These triangles are similar because

150 hapter 3. SIMILRITY 397. onstruct a triangle, given the ratio of its altitude to the base, the angle at the vertex, and the median drawn to one of its lateral sides 398. Into a given disk segment,

More information

### 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

More information

### 10-4 Inscribed Angles. Find each measure. 1.

Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semi-circle. 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 information

### circle 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 information

### 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.

More information

### 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

More information

### Test 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 information

### CONJECTURES - Discovering Geometry. Chapter 2

CONJECTURES - Discovering Geometry Chapter C-1 Linear Pair Conjecture - If two angles form a linear pair, then the measures of the angles add up to 180. C- Vertical Angles Conjecture - If two angles are

More information

### 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

More information

### Dates, assignments, and quizzes subject to change without advance notice. Monday Tuesday Block Day Friday. 3 (only see 6 th, 4

Name: Period GL UNIT 12: IRLS I can define, identify and illustrate the following terms: Interior of a circle hord xterior of a circle Secant of a circle Tangent to a circle Point of tangency entral angle

More information

### 10.5. Angle Relationships in Circles For use with Exploration 10.5. Essential Question When a chord intersects a tangent line or another

Name ate 0.5 ngle Relationships in ircles For use with Exploration 0.5 Essential Question When a chord intersects a tangent line or another chord, what relationships exist among the angles and arcs formed?

More information

### Chapter 4 Circles, Tangent-Chord Theorem, Intersecting Chord Theorem and Tangent-secant Theorem

Tampines Junior ollege H3 Mathematics (9810) Plane Geometry hapter 4 ircles, Tangent-hord Theorem, Intersecting hord Theorem and Tangent-secant Theorem utline asic definitions and facts on circles The

More information

### 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

More information

### Final Review Problems Geometry AC Name

Final Review Problems Geometry Name SI GEOMETRY N TRINGLES 1. The measure of the angles of a triangle are x, 2x+6 and 3x-6. Find the measure of the angles. State the theorem(s) that support your equation.

More information

### (a) 5 square units. (b) 12 square units. (c) 5 3 square units. 3 square units. (d) 6. (e) 16 square units

1. Find the area of parallelogram ACD shown below if the measures of segments A, C, and DE are 6 units, 2 units, and 1 unit respectively and AED is a right angle. (a) 5 square units (b) 12 square units

More information

### 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,

More information

### 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.

More information

### A = ½ x b x h or ½bh or bh. Formula Key A 2 + B 2 = C 2. Pythagorean Theorem. Perimeter. b or (b 1 / b 2 for a trapezoid) height

Formula Key b 1 base height rea b or (b 1 / b for a trapezoid) h b Perimeter diagonal P d (d 1 / d for a kite) d 1 d Perpendicular two lines form a angle. Perimeter P = total of all sides (side + side

More information

### 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..................................

More information

### Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade

Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade Standards/Content Padrões / Conteúdo Learning Objectives Objetivos de Aprendizado Vocabulary Vocabulário Assessments Avaliações Resources

More information

### 12-1. Tangent Lines. Vocabulary. Review. Vocabulary Builder HSM11_GEMC_1201_T Use Your Vocabulary

1-1 Tangent Lines Vocabulary Review 1. ross out the word that does NT apply to a circle. arc circumference diameter equilateral radius. ircle the word for a segment with one endpoint at the center of a

More information

### Geometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles

Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.

More information

### Duplicating Segments and Angles

ONDENSED LESSON 3.1 Duplicating Segments and ngles In this lesson you will Learn what it means to create a geometric construction Duplicate a segment by using a straightedge and a compass and by using

More information

### Name Geometry Exam Review #1: Constructions and Vocab

Name Geometry Exam Review #1: Constructions and Vocab Copy an angle: 1. Place your compass on A, make any arc. Label the intersections of the arc and the sides of the angle B and C. 2. Compass on A, make

More information

### 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

More information

### 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

More information

### Overview Mathematical Practices Congruence

Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason

More information

### 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

More information

### 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,

More information

### The 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 information

### 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,

More information

### Lesson 3.1 Duplicating Segments and Angles

Lesson 3.1 Duplicating Segments and ngles In Exercises 1 3, use the segments and angles below. Q R S 1. Using only a compass and straightedge, duplicate each segment and angle. There is an arc in each

More information

### A. 3y = -2x + 1. y = x + 3. y = x - 3. D. 2y = 3x + 3

Name: Geometry Regents Prep Spring 2010 Assignment 1. Which is an equation of the line that passes through the point (1, 4) and has a slope of 3? A. y = 3x + 4 B. y = x + 4 C. y = 3x - 1 D. y = 3x + 1

More information

### Sec 1.1 CC Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB.

Sec 1.1 CC Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB. A B C **Using a ruler measure the two lengths to make sure they have

More information

### 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

More information

### Basics of Circles 9/20/15. Important theorems:

Basics of ircles 9/20/15 B H P E F G ID Important theorems: 1. A radius is perpendicular to a tangent at the point of tangency. PB B 2. The measure of a central angle is equal to the measure of its intercepted

More information

### 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

More information

### 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

More information

### Geometry: 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 information

### 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.

More information

### 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

More information

### Geometry 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 information

### Radius, 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 information

### 1. A person has 78 feet of fencing to make a rectangular garden. What dimensions will use all the fencing with the greatest area?

1. A person has 78 feet of fencing to make a rectangular garden. What dimensions will use all the fencing with the greatest area? (a) 20 ft x 19 ft (b) 21 ft x 18 ft (c) 22 ft x 17 ft 2. Which conditional

More information

### 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,

More information

### 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,

More information

### 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

More information

### Circle theorems CHAPTER 19

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

More information

### 2006 ACTM STATE GEOMETRY EXAM

2006 TM STT GOMTRY XM In each of the following you are to choose the best (most correct) answer and mark the corresponding letter on the answer sheet provided. The figures are not necessarily drawn to

More information

### 0810ge. Geometry Regents Exam 0810

0810ge 1 In the diagram below, ABC XYZ. 3 In the diagram below, the vertices of DEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ABC is 36 cm. Which two statements identify

More information

### Duplicating 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 information

### Chapters 4 and 5 Notes: Quadrilaterals and Similar Triangles

Chapters 4 and 5 Notes: Quadrilaterals and Similar Triangles IMPORTANT TERMS AND DEFINITIONS parallelogram rectangle square rhombus A quadrilateral is a polygon that has four sides. A parallelogram is

More information

### Chapter 1 Exam. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. 1.

Name: lass: ate: I: hapter 1 Exam Multiple hoice Identify the choice that best completes the statement or answers the question. 1. bisects, m = (7x 1), and m = (4x + 8). Find m. a. m = c. m = 40 b. m =

More information

### 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

More information

### Geometry Vocabulary. Created by Dani Krejci referencing:

Geometry Vocabulary Created by Dani Krejci referencing: http://mrsdell.org/geometry/vocabulary.html point An exact location in space, usually represented by a dot. A This is point A. line A straight path

More information

### Honors Geometry Final Exam Study Guide

2011-2012 Honors Geometry Final Exam Study Guide Multiple Choice Identify the choice that best completes the statement or answers the question. 1. In each pair of triangles, parts are congruent as marked.

More information

### 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

More information

### Chapter Three. Circles

hapter Three ircles FINITIONS 1. QUL IRLS are those with equal radii. K 2. HOR of a circle is any straight line joining two points on its circumference. For example, K is a chord. R 3. n R of a circle

More information

### INDEX. Arc Addition Postulate,

# 30-60 right triangle, 441-442, 684 A Absolute value, 59 Acute angle, 77, 669 Acute triangle, 178 Addition Property of Equality, 86 Addition Property of Inequality, 258 Adjacent angle, 109, 669 Adjacent

More information

### 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

More information

### An inscribed polygon is a polygon whose vertices are points of a circle. in means in and scribed means written

Geometry Week 6 Sec 3.5 to 4.1 Definitions: section 3.5 A A B D B D C Connect the points in consecutive order with segments. C The square is inscribed in the circle. An inscribed polygon is a polygon whose

More information

### Inscribed 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 information

### 1) Perpendicular bisector 2) Angle bisector of a line segment

1) Perpendicular bisector 2) ngle bisector of a line segment 3) line parallel to a given line through a point not on the line by copying a corresponding angle. 1 line perpendicular to a given line through

More information

### 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

More information

### EUCLIDEAN 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 information

### Chapter 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 information