# Lesson 1: Introducing Circles

Size: px
Start display at page:

## Transcription

1 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 angles, radii, and chords of a circle? 3. What are the relationships among circumscribed angles, central angles, and inscribed angles? 4. What is the relationship between a tangent line and the radius of a circle? WORS TO KNOW arc central angle chord circle circumference circumscribed angle concentric circles congruent arcs part of a circle s circumference an angle with its vertex at the center of a circle a segment whose endpoints lie on the circumference of the circle the set of all points that are equidistant from a reference point, the center. The set of points forms a 2-dimensional curve that measures 360º. the distance around a circle; = 2 r or = d, for which represents circumference, r represents the circle s radius, and d represents the circle s diameter. the angle formed by two tangent lines whose vertex is outside of the circle coplanar circles that have the same center two arcs that have the same measure and are either of the same circle or of congruent circles U3-1 Lesson 1: Introducing ircles

2 diameter inscribed angle intercepted arc major arc minor arc pi ( ) radius secant line semicircle tangent line a straight line passing through the center of a circle connecting two points on the circle; equal to twice the radius an angle formed by two chords whose vertex is on the circle an arc whose endpoints intersect the sides of an inscribed angle and whose other points are in the interior of the angle part of a circle s circumference that is larger than its semicircle part of a circle s circumference that is smaller than its semicircle the ratio of circumference of a circle to the diameter; equal to approximately 3.14 the distance from the center to a point on the circle; equal to one-half the diameter a line that intersects a circle at two points an arc that is half of a circle a line that intersects a circle at exactly one point and is perpendicular to the radius of the circle U3-2 Unit 3: ircles and Volume

3 Recommended Resources Math Open Reference. entral ngle Theorem. This site describes the entral ngle Theorem and allows users to explore the relationship between inscribed angles and central angles. Math Warehouse. What Is the Tangent of a ircle? This site reviews the properties of tangent lines, and allows users to interactively explore the idea that a tangent line is perpendicular to a radius at the point of tangency. This site also provides limited practice problems, as well as solutions. RicksMath.com. Geometry and the ircle. This site defines and provides examples of relationships of circles and their properties. U3-3 Lesson 1: Introducing ircles

4 Lesson 3.1.1: Similar ircles and entral and Inscribed ngles Introduction In the third century.., Greek mathematician Euclid, often referred to as the Father of Geometry, created what is known as Euclidean geometry. He took properties of shape, size, and space and postulated their unchanging relationships that cultures before understood but had not proved to always be true. rchimedes, a fellow Greek mathematician, followed that by creating the foundations for what is now known as calculus. In addition to being responsible for determining things like the area under a curve, rchimedes is credited for coming up with a method for determining the most accurate approximation of pi,. In this lesson, you will explore and practice applying several properties of circles including proving that all circles are similar using a variation of rchimedes method. Key oncepts Pi, ( ), is the ratio of the circumference to the diameter of a circle, where the circumference is the distance around a circle, the diameter is a segment with endpoints on the circle that passes through the center of the circle, and a circle is the set of all points that are equidistant from a reference point (the center) and form a 2-dimensional curve. circle measures 360º. oncentric circles share the same center. The diagram below shows circle ( ) with diameter and radius. The radius of a circle is a segment with endpoints on the circle and at the circle s center; a radius is equal to half the diameter. radius diameter ll circles are similar and measure 360º. U3-4 Unit 3: ircles and Volume

5 portion of a circle s circumference is called an arc. The measure of a semicircle, or an arc that is equal to half of a circle, is 180º. rcs are named by their endpoints. The semicircle in the following diagram can be named. m = 180 ) part of the circle that is larger than a semicircle is called a major arc. It is common to identify a third point on the circle when naming major arcs. The major arc in the following diagram can be named. minor arc is a part of a circle that is smaller than a semicircle. U3-5 Lesson 1: Introducing ircles

6 The minor arc in the following diagram can be named. Two arcs of the same circle or of congruent circles are congruent arcs if they have the same measure. The measure of an arc is determined by the central angle. central angle of a circle is an angle with its vertex at the center of the circle and sides that are created from two radii of the circle. chord is a segment whose endpoints lie on the circumference of a circle. n inscribed angle of a circle is an angle formed by two chords whose vertex is on the circle. U3-6 Unit 3: ircles and Volume

7 n inscribed angle is half the measure of the central angle that intercepts the same arc. onversely, the measure of the central angle is twice the measure of the inscribed angle that intercepts the same arc. This is called the Inscribed ngle Theorem. Inscribed ngle Theorem The measure of an inscribed angle is half the measure of its intercepted arc s angle. Given, m = 1 m. 2 In the following diagram, is the inscribed angle and is the central angle. They both intercept the minor arc. 2x x U3-7 Lesson 1: Introducing ircles

8 orollaries to the Inscribed ngle Theorem orollary 1 orollary 2 Two inscribed angles that intercept the same arc are congruent. n angle inscribed in a semicircle is a right angle. E U3-8 Unit 3: ircles and Volume

9 Guided Practice Example 1 Prove that the measure of a central angle is twice the measure of an inscribed angle that intercepts the same arc. Given: with inscribed and central intercepting. Prove: 2m = m 1. Identify the known information. ircle with inscribed and central intercepts. Let be a diameter of the circle. y definition, and are radii of the circle. Mark them as r. Identify as x and as y. r y r x U3-9 Lesson 1: Introducing ircles

10 2. Identify what information is known about the angles of the triangle. is isosceles since both legs are the same length, r. y the Isosceles Triangle Theorem, both base angles of the triangle are congruent; therefore, = x. lso, the vertex angle of the isosceles triangle is a linear pair with y and thus m = 180 y. x r y r x y the Triangle Sum Theorem: x + x y = 180 2x y = 0 2x = y Substituting the names for x and y yields 2m = m. Therefore, the measure of a central angle is twice the measure of an inscribed angle that intercepts the same arc. U3-10 Unit 3: ircles and Volume

11 Example 2 Prove that all circles are similar using the concept of similarity transformations. 1. reate a diagram to help with the proof. raw with radius r 1. r 1 raw with radius r 2 such that r 2 > r 1. r 2 U3-11 Lesson 1: Introducing ircles

12 2. Using properties of rigid motion, translate so that it is concentric with. r 2 r 1 3. etermine the scale factor necessary to dilate so that it maps to. To determine the ratio, note that each circle has a radius that by definition is the segment from the center to a point on the circle. To map, divide r 2 by r 1. The resulting ratio is the scale factor that produces the following image. r 2 r 1 ' This scale factor, r 2, is true for all circles and a similarity r1 transformation of this sort can always be used because all circles are similar. U3-12 Unit 3: ircles and Volume

13 Example 3 car has a circular turning radius of 15.5 feet. The distance between the two front tires is 5.4 feet. To the nearest foot, how much farther does a tire on the outer edge of the turning radius travel than a tire on the inner edge? 15.5 ft 5.4 ft 1. alculate the circumference of the outer tire s turn. = 2πr Formula for the circumference of a circle = 2π ( 155. ) =31π Substitute 15.5 for the radius (r). Simplify. 2. alculate the circumference of the inside tire s turn. First, calculate the radius of the inner tire s turn. Since all tires are similar, the radius of the inner tire s turn can be calculated by subtracting the distance between the two front wheels (the distance between each circle) from the radius of the outer tire s turn = 10.1 ft = 2πr Formula for the circumference of a circle = 2π ( 101. ) =202. π Simplify. Substitute 10.1 for the radius (r). U3-13 Lesson 1: Introducing ircles

14 3. alculate the difference in the circumference of each tire s turn. Find the difference in the circumference of each tire s turn. 31π 202. π= 108. π The outer tire travels approximately 34 feet farther than the inner tire. Example 4 Find the value of each variable. a E b c F 1. Identify the inscribed angle and the intercepted arc and use the proper theorem or corollary to find the value of a. a is the inscribed angle creating the intercepting arc. The measure of the intercepted arc is 70º. Use the Inscribed ngle Theorem to find a. m E= m 2 Inscribed ngle Theorem 1 a= ( 70) 2 Substitute the given information. a = 35 Solve for a. The value of a is 35º. U3-14 Unit 3: ircles and Volume

15 2. Identify the inscribed angle and the intercepted arc and use the proper theorem or corollary to find the value of b. b is the inscribed angle creating the intercepting arc EF. The measure of the intercepted arc is 104º. Use the Inscribed ngle Theorem to find b. m = mef 2 Inscribed ngle Theorem 1 b= ( 104) 2 Substitute the given information. b = 52 Solve for b. The value of b is 52º. 3. Use the Triangle Sum Theorem and the Vertical ngles Theorem to find the value of c. Note that a and b, in addition to being the measures of inscribed angles, are interior angles of a triangle. The third interior angle is a vertical angle to c and is therefore congruent to c. Finding that third angle will yield the value of c. a + b + c = 180 Triangle Sum Theorem and Vertical ngles Theorem c = 180 Substitute the measures of a and b c = 180 ombine like terms. c = 93 Solve for c. The value of c is 93º. U3-15 Lesson 1: Introducing ircles

16 Example 5 Find the measures of and. (7x 7) (x + 14) 1. Set up an equation to solve for x. is a central angle and is an inscribed angle in. m = 2m entral ngle/inscribed ngle Theorem 7x 7 = 2(x + 14) Substitute values for and. 7x 7 = 2x + 28 istributive Property 5x = 35 Solve for x. x = 7 2. Substitute the value of x into the expression for to find the measure of the inscribed angle. m = ( x+ 14 ) m = ( 7) + 14 m =21 The measure of is 21º. U3-16 Unit 3: ircles and Volume

17 3. Find the value of the central angle,. y the Inscribed ngle Theorem, m = 2 m. m =21( 2) m =42 The measure of is 42º. U3-17 Lesson 1: Introducing ircles

18 PRTIE IRLES N VOLUME Lesson 1: Introducing ircles Practice 3.1.1: Similar ircles and entral and Inscribed ngles Given that all circles are similar, determine the scale factor necessary to map. 1. has a radius of 144 units and has a radius of 3 units. 2. has a diameter of 50 units and has a diameter of 12 units. Use your knowledge of similar circles to complete problems 3 and circular waterfall is surrounded by a brick wall. The radius of the inner wall is 35 inches. If the bricks are 5 inches wide, what scale factor was used to determine the radius of the outer wall? 35 in 5 in 4. small car has a tire with a 17-inch diameter. truck has a tire with a 32-inch diameter. How much farther than the car does the truck have to drive for its tire to complete one revolution? U3-18 Unit 3: ircles and Volume continued

19 PRTIE IRLES N VOLUME Lesson 1: Introducing ircles Use your knowledge of angles to complete the problems that follow. 5. Find the values of x, y, and z. x z y 6. Find the value of x and the measure of. (x + 7) (3x 5) continued U3-19 Lesson 1: Introducing ircles

20 PRTIE IRLES N VOLUME Lesson 1: Introducing ircles 7. Find the values of x and y. 15 y x 8. Find m and m. ( x + 5) 3 (x 35) continued U3-20 Unit 3: ircles and Volume

21 PRTIE IRLES N VOLUME Lesson 1: Introducing ircles 9. Find m and m. y 100 (2x + 10) 10. Find m and m U3-21 Lesson 1: Introducing ircles

22 Lesson 3.1.2: hord entral ngles onjecture Introduction ircles have several special properties, conjectures, postulates, and theorems associated with them. This lesson focuses on the relationship between chords and the angles and arcs they create. Key oncepts hords are segments whose endpoints lie on the circumference of a circle. Three chords are shown on the circle below. ongruent chords of a circle create one pair of congruent central angles. U3-22 Unit 3: ircles and Volume

23 When the sides of the central angles create diameters of the circle, vertical angles are formed. This creates two pairs of congruent central angles. E ongruent chords also intercept congruent arcs. n intercepted arc is an arc whose endpoints intersect the sides of an inscribed angle and whose other points are in the interior of the angle. Remember that the measure of an arc is the same as the measure of its central angle. lso, recall that central angles are twice the measure of their inscribed angles. entral angles of two different triangles are congruent if their chords and circles are congruent. In the circle below, chords and E are congruent chords of. E U3-23 Lesson 1: Introducing ircles

24 When the radii are constructed such that each endpoint of the chord connects to the center of the circle, four central angles are created, as well as two congruent isosceles triangles by the SSS ongruence Postulate. E Since the triangles are congruent and both triangles include two central angles that are the vertex angles of the isosceles triangles, those central angles are also congruent because orresponding Parts of ongruent Triangles are ongruent (PT). E The measure of the arcs intercepted by the chords is congruent to the measure of the central angle because arc measures are determined by their central angle. x x x x E U3-24 Unit 3: ircles and Volume

25 Guided Practice Example 1 In, m =57. What is m? Find the measure of. m =57 The measure of is equal to the measure of because central angles are congruent to their intercepted arc; therefore, the measure of is also 57º. 2. Find the measure of. Subtract the measure of from 360º = 303 m State your conclusion. The measure of is 303º. U3-25 Lesson 1: Introducing ircles

26 Example 2 G E. What conclusions can you make? K G I E J H 1. What type of angles are G and E, and what does that tell you about the chords? G and E are congruent central angles of congruent circles, so HI and JK are congruent chords. 2. Since the central angles are congruent, what else do you know? The measures of the arcs intercepted by congruent chords are congruent, so minor arc HI is congruent to minor arc JK. eductively, then, major arc IH is congruent to major arc KJ. U3-26 Unit 3: ircles and Volume

27 Example 3 Find the value of y. (13y 48) (8y 47) E 1. Set up an equation to solve for y. The angles marked are central angles created by congruent chords, and are therefore congruent. Set the measure of each angle equal to each other. 13y 48 = 8y Solve for y. 13y 48 = 8y y = 95 y = 19 entral angles created by congruent chords are congruent. U3-27 Lesson 1: Introducing ircles

28 PRTIE IRLES N VOLUME Lesson 1: Introducing ircles Practice 3.1.2: hord entral ngles onjecture Use what you ve learned about chords and central angles to solve. 1. In, m =72. What is m? In, 315. What is m? What is the value of w? (43 3w) (w + 9) continued U3-28 Unit 3: ircles and Volume

29 PRTIE IRLES N VOLUME Lesson 1: Introducing ircles 4. What can you conclude about G and E? K G I E J H 5. G E. What is the value of a? K 3a G (6a 24) I E J H 6. Find the value of b. (5b 9) (8 2b) E continued U3-29 Lesson 1: Introducing ircles

30 PRTIE IRLES N VOLUME Lesson 1: Introducing ircles 7. Find the value of b. (8b + 1) (12 + 6b) E 8. Find the value of b. (8b + 11) (10b + 3) E 9. If the circumference of one circle is 54 millimeters and the major arc of a second circle measures 182º, what is the length of this major arc of the second circle? 10. You re sewing the seam on a circular lampshade for interior design class. The total circumference of the lampshade is 3 feet. The amount of the seam you ve sewn so far measures approximately 23º around the circumference of the lampshade. What length of lampshade still needs to be sewn? U3-30 Unit 3: ircles and Volume

31 Lesson 3.1.3: Properties of Tangents of a ircle Introduction ircles and tangent lines can be useful in many real-world applications and fields of study, such as construction, landscaping, and engineering. There are many different types of lines that touch or intersect circles. ll of these lines have unique properties and relationships to a circle. Specifically, in this lesson, we will identify what a tangent line is, explore the properties of tangent lines, prove that a line is tangent to a circle, and find the lengths of tangent lines. We will also identify and use secant lines, as well as discuss how they are different from tangent lines. Key oncepts tangent line is a line that intersects a circle at exactly one point. Tangent lines are perpendicular to the radius of the circle at the point of tangency. Point of tangency Radius Tangent line You can verify that a line is tangent to a circle by constructing a right triangle using the radius, and verifying that it is a right triangle by using the Pythagorean Theorem. The slopes of a line and a radius drawn to the possible point of tangency must be negative reciprocals in order for the line to be a tangent. If two segments are tangent to the same circle, and originate from the same exterior point, then the segments are congruent. The angle formed by two tangent lines whose vertex is outside of the circle is called the circumscribed angle. U3-31 Lesson 1: Introducing ircles

32 is a circumscribed angle. O The length of tangent is equal to the length of tangent. The angle formed by two tangents is equal to one half the positive difference of the angle s intercepted arcs. secant line is a line that intersects a circle at two points. Secant line n angle formed by a secant and a tangent is equal to the positive difference of its intercepted arcs. U3-32 Unit 3: ircles and Volume

33 Guided Practice Example 1 etermine whether is tangent to in the diagram below Identify the radius. The radius of the circle is the segment from the center of the circle to a point on the circle. is the radius of. 2. etermine the relationship between and at point in order for to be tangent to. must be perpendicular to at point in order for to be tangent to. U3-33 Lesson 1: Introducing ircles

34 3. Show that is a right angle by using the converse of the Pythagorean Theorem. The converse of the Pythagorean Theorem states that when the sum of the squares of two sides of a triangle is equal to the square of the third side of the triangle, the triangle is a right triangle. a 2 + b 2 = c 2 Pythagorean Theorem () 2 + () 2 = () 2 Substitute segment names for a, b, and c = 41 2 Substitute values for,, and = 1681 Simplify, then solve = 1681 The result is a true statement; therefore, is a right angle. 4. State your conclusion. Since the converse of the Pythagorean Theorem is true, is a right triangle; therefore, is a right angle. This makes perpendicular to ; therefore, is tangent to. U3-34 Unit 3: ircles and Volume

35 Example 2 Each side of is tangent to circle O at the points, E, and F. Find the perimeter of. 7 F O 5 E Identify the lengths of each side of the triangle. is tangent to the same circle as F and extends from the same point; therefore, the lengths are equal. = 7 units E is tangent to the same circle as and extends from the same point; therefore, the lengths are equal. E = 5 units To determine the length of E, subtract the length of E from the length of = 11 E = 11 units F is tangent to the same circle as E and extends from the same point; therefore, the lengths are equal. F = 11 units U3-35 Lesson 1: Introducing ircles

36 2. alculate the perimeter of. dd the lengths of, F,, E, E, and F to find the perimeter of the polygon = 46 units The perimeter of is 46 units. Example 3 landscaper wants to build a walkway tangent to the circular park shown in the diagram below. The other walkway pictured is a radius of the circle and has a slope of 1 on the grid. If the walkways should intersect at (4, 2) on the grid, what equation 2 can the landscaper use to graph the new walkway on the grid? y x U3-36 Unit 3: ircles and Volume

37 1. etermine the slope of the walkway. One of the properties of tangent lines states that if a line is tangent to a circle, then it is perpendicular to the radius at the point of tangency. Since the slope of the radius is 1, then the slope of the new walkway will 2 be 2 because perpendicular lines have slopes that are negative reciprocals. 2. etermine the equation that represents the walkway. Recall that the point-slope form of a line is y y = m( x x ). 1 1 Since the new walkway is tangent to the circle at (4, 2), then the point (4, 2) is on the tangent line, so it can be used to write the equation. y ( 2) = 2( x 4) y+ 2= 2( x 4) This equation can be rearranged into slope-intercept form, y= 2x 10, for easier graphing. Example 4 is tangent to at point as shown below. Find the length of as well as m U3-37 Lesson 1: Introducing ircles

38 1. Find the length of. Since is tangent to, then is right angle because a tangent and a radius form a right angle at the point of tangency. Since is a right angle, is a right triangle. Use the Pythagorean Theorem to find the length of. a 2 + b 2 = c 2 Pythagorean Theorem () 2 = 17 2 Substitute values for a, b, and c () 2 = 289 Simplify. () 2 = 225 = 15 The length of is 15 units. 2. Find m. First, determine the unknown measure of. Recall that the sum of all three angles of a triangle is 180º. is a right angle, so it is 90º. is 28º, as shown in the diagram. Set up an equation to determine the measure of m = m = 180 m = 62 Since m = 62, then m = 118 because and are a linear pair. is a central angle, and recall that the measure of a central angle is the same as its intercepted arc, so is 118º. U3-38 Unit 3: ircles and Volume

39 PRTIE IRLES N VOLUME Lesson 1: Introducing ircles Practice 3.1.3: Properties of Tangents of a ircle Use what you have learned about tangent lines and secant lines to answer the questions. 1. and are tangent to L in the diagram below. What is the value of x? (3x 10) L (x + 6) 2. You know that PR is tangent to S in the diagram below. What must you prove to show that this is true? Q P S R continued U3-39 Lesson 1: Introducing ircles

40 PRTIE IRLES N VOLUME Lesson 1: Introducing ircles 3. is tangent to at point in the diagram below. What is the measure of? space station orbiting the Earth is sending two signals that are tangent to the Earth. If the intercepted arc of the Earth s surface that is visible to the satellite is 168º, what is the measure of the angle formed by the two signal beams? 5. Is XY tangent to Z at point X in the diagram below? Explain. X 55 Y Z continued U3-40 Unit 3: ircles and Volume

41 PRTIE IRLES N VOLUME Lesson 1: Introducing ircles 6. How many tangents can be drawn that contain a point on a circle? Explain your answer. 7. The slope of radius PQ in circle Q is 2. student wants to draw a tangent to 3 Q at point P. What will be the slope of this tangent line? 8. homeowner is building a square fence around his circular patio so that each side of the fence is tangent to the circle, as shown in the diagram below. The radius of his patio is 18 yards. What is the perimeter of his fence? continued U3-41 Lesson 1: Introducing ircles

42 PRTIE IRLES N VOLUME Lesson 1: Introducing ircles 9. You are using binoculars to look at an eagle. The sides of the binoculars seem to extend from the eagle, and are tangent to the circular portion. How far away is the eagle? Eagle (x + 51) ft (18x) ft 10. Your friend Truman argues that chords and tangents are the same thing. You disagree. What do you tell Truman to convince him that he is incorrect? U3-42 Unit 3: ircles and Volume

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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,

### 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.

### 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

### 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

### 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,

### 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

### 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

### 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

### 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

### 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

### 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.

### 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

### 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

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

### 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

### 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

### 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

### 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,

### 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

### 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

### 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

### 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,

### 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

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### 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:

### 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

### 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

### 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

### 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

### 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

### 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)

### 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. 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

### Geometry Unit 6 Areas and Perimeters

Geometry Unit 6 Areas and Perimeters Name Lesson 8.1: Areas of Rectangle (and Square) and Parallelograms How do we measure areas? Area is measured in square units. The type of the square unit you choose

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### 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.

### 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 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

### Unit 7 Circles. Vocabulary and Formulas for Circles:

ccelerated G Unit 7 ircles Name & ate Vocabulary and Formulas for ircles: irections: onsider 1) Find the circumference of the circle. to answer the following questions. Exact: pproximate: 2) Find the area

### 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

### Arc Length and Areas of Sectors

Student Outcomes When students are provided with the angle measure of the arc and the length of the radius of the circle, they understand how to determine the length of an arc and the area of a sector.

### Surface Area and Volume Cylinders, Cones, and Spheres

Surface Area and Volume Cylinders, Cones, and Spheres Michael Fauteux Rosamaria Zapata CK12 Editor Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable

### Circumference CHAPTER. www.ck12.org 1

www.ck12.org 1 CHAPTER 1 Circumference Here you ll learn how to find the distance around, or the circumference of, a circle. What if you were given the radius or diameter of a circle? How could you find

### 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,

### Teacher Page Key. Geometry / Day # 13 Composite Figures 45 Min.

Teacher Page Key Geometry / Day # 13 Composite Figures 45 Min. 9-1.G.1. Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles

### Student Outcomes. Lesson Notes. Classwork. Exercises 1 3 (4 minutes)

Student Outcomes Students give an informal derivation of the relationship between the circumference and area of a circle. Students know the formula for the area of a circle and use it to solve problems.

### Additional Topics in Math

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

### GEOMETRIC MENSURATION

GEOMETRI MENSURTION Question 1 (**) 8 cm 6 cm θ 6 cm O The figure above shows a circular sector O, subtending an angle of θ radians at its centre O. The radius of the sector is 6 cm and the length of the

### 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. 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

### Week 1 Chapter 1: Fundamentals of Geometry. Week 2 Chapter 1: Fundamentals of Geometry. Week 3 Chapter 1: Fundamentals of Geometry Chapter 1 Test

Thinkwell s Homeschool Geometry Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Geometry! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson plan

### 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

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### GAP CLOSING. 2D Measurement. Intermediate / Senior Student Book

GAP CLOSING 2D Measurement Intermediate / Senior Student Book 2-D Measurement Diagnostic...3 Areas of Parallelograms, Triangles, and Trapezoids...6 Areas of Composite Shapes...14 Circumferences and Areas

### Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes

### Lesson 19: Equations for Tangent Lines to Circles

Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line

### INTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE

INTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE ABSTRACT:- Vignesh Palani University of Minnesota - Twin cities e-mail address - palan019@umn.edu In this brief work, the existing formulae

### Right Triangles 4 A = 144 A = 16 12 5 A = 64

Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right

### 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

### Lesson 9.1 The Theorem of Pythagoras

Lesson 9.1 The Theorem of Pythagoras Give all answers rounded to the nearest 0.1 unit. 1. a. p. a 75 cm 14 cm p 6 7 cm 8 cm 1 cm 4 6 4. rea 9 in 5. Find the area. 6. Find the coordinates of h and the radius

### 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

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### Applications for Triangles

Not drawn to scale Applications for Triangles 1. 36 in. 40 in. 33 in. 1188 in. 2 69 in. 2 138 in. 2 1440 in. 2 2. 188 in. 2 278 in. 2 322 in. 2 none of these Find the area of a parallelogram with the given

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

Name: lass: _ ate: _ I: SSS Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Given the lengths marked on the figure and that bisects E, use SSS to explain

### 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

### 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.

### Inv 1 5. Draw 2 different shapes, each with an area of 15 square units and perimeter of 16 units.

Covering and Surrounding: Homework Examples from ACE Investigation 1: Questions 5, 8, 21 Investigation 2: Questions 6, 7, 11, 27 Investigation 3: Questions 6, 8, 11 Investigation 5: Questions 15, 26 ACE

### CHAPTER 8, GEOMETRY. 4. A circular cylinder has a circumference of 33 in. Use 22 as the approximate value of π and find the radius of this cylinder.

TEST A CHAPTER 8, GEOMETRY 1. A rectangular plot of ground is to be enclosed with 180 yd of fencing. If the plot is twice as long as it is wide, what are its dimensions? 2. A 4 cm by 6 cm rectangle has

### Geometry EOC Practice Test #2

Class: Date: Geometry EOC Practice Test #2 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Rebecca is loading medical supply boxes into a crate. Each supply

### 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

### 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

### Sample Problems. Practice Problems

Lecture Notes Circles - Part page Sample Problems. Find an equation for the circle centered at (; ) with radius r = units.. Graph the equation + + = ( ).. Consider the circle ( ) + ( + ) =. Find all points

### 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

### 12 Surface Area and Volume

12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

### 43 Perimeter and Area

43 Perimeter and Area Perimeters of figures are encountered in real life situations. For example, one might want to know what length of fence will enclose a rectangular field. In this section we will study

### G C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Performance Assessment Task Circle and Squares Grade 10 This task challenges a student to analyze characteristics of 2 dimensional shapes to develop mathematical arguments about geometric relationships.

### Chapter 4.1 Parallel Lines and Planes

Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about

### 1. A plane passes through the apex (top point) of a cone and then through its base. What geometric figure will be formed from this intersection?

Student Name: Teacher: Date: District: Description: Miami-Dade County Public Schools Geometry Topic 7: 3-Dimensional Shapes 1. A plane passes through the apex (top point) of a cone and then through its

### 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,

### 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 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

### Tangent circles in the hyperbolic disk

Rose- Hulman Undergraduate Mathematics Journal Tangent circles in the hyperbolic disk Megan Ternes a Volume 14, No. 1, Spring 2013 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics

### 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

### Lesson 18: Looking More Carefully at Parallel Lines

Student Outcomes Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180. They learn how to prove the alternate interior angles theorem using

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### 7.4A/7.4B STUDENT ACTIVITY #1

7.4A/7.4B STUDENT ACTIVITY #1 Write a formula that could be used to find the radius of a circle, r, given the circumference of the circle, C. The formula in the Grade 7 Mathematics Chart that relates the

### High School Geometry Test Sampler Math Common Core Sampler Test

High School Geometry Test Sampler Math Common Core Sampler Test Our High School Geometry sampler covers the twenty most common questions that we see targeted for this level. For complete tests and break

### 16 Circles and Cylinders

16 Circles and Cylinders 16.1 Introduction to Circles In this section we consider the circle, looking at drawing circles and at the lines that split circles into different parts. A chord joins any two

### 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