# The Geometry of a Circle Geometry (Grades 10 or 11)

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 The Geometry of a Circle Geometry (Grades 10 or 11) A 5 day Unit Plan using Geometers Sketchpad, graphing calculators, and various manipulatives (string, cardboard circles, Mira s, etc.). Dennis Kapatos I2T2 Project 12/1/05

2 Unit Overview Unit Objectives: Students will learn a broad range of skills and content knowledge. In addition to all the theorems in each section, students will be able to make observations and conjectures and to test these conjectures using the technology and manipulatives at their disposal. Students will also be able to work cooperatively with other group members to investigate the properties of geometric figures (circles more specifically) and prove theorems. In addition to these, students will be able to recognize applications of circles and their related parts in the world around them. NCTM Standards: Number and Operation Students judge the reasonableness of numerical computations and their results. Algebra Students draw reasonable conclusions about a situation being modeled. Geometry Students explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them. Students establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others. Students use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations. Students use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest. Measurement Students make decisions about units and scales that are appropriate for problem situations involving measurement. Problem Solving Students build new mathematical knowledge through problem solving. Students solve problems that arise in mathematics and in other contexts. Reasoning and Proof Students make and investigate mathematical conjectures. Students develop and evaluate mathematical arguments and proofs. Students select and use various types of reasoning and methods of proof. Communication Students organize and consolidate their mathematical thinking through communication. Connections Students recognize and apply mathematics in contexts outside of mathematics. New York State Standards: G.PS.6 Use a variety of strategies to extend solution methods to other problems. G.PS.8 Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions. G.CM.5 Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid.

3 G.CN.7 Recognize and apply mathematical ideas to problem situations that develop outside of mathematics. G.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts. G.R.3 Use representation as a tool for exploring and understanding mathematical ideas. G.G.27 Write a proof arguing from a given hypothesis to a given conclusion. G.G.29 Identify corresponding parts of congruent triangles. G.G.49 Investigate, justify, and apply theorems regarding chords of a circle. G.G.50 Investigate, justify, and apply theorems about tangent lines to a circle. G.G.51 Investigate, justify, and apply theorems about the arcs determined by the rays of angles formed by two lines intersecting a circle. G.G.52 Investigate, justify, and apply theorems about arcs of a circle cut by two parallel lines. G.G.53 Investigate, justify, and apply theorems regarding segments intersected by a circle. Materials and Equipment: Resources: Geometer s Sketchpad Computers Projector Graphing Calculators Compasses Mira s Rulers Protractors String Cardboard Circles Empty Cans Textbook: New York Math A/B: An Integrated Approach Volume 2 Bass, Hall, Johnson, and Wood. New York Math A/B: An Integrated Approach Volume 2. Teacher s Edition. Prentice Hall, Chapter 12. Pages Bennett, Dan. Exploring Geometry with The Geometer s Sketchpad. 4 th Edition. Key Curriculum Press, Chapter 6. Pages Unit Description: This unit is designed to cover chapter 12 sections one through five. Although the chapter has 6 sections, this remaining section could be covered similar to the previous five. The five lessons are intended to be inquiry based, though some of the teacher s instructions may not seem so. The teacher should give only as much help as is needed to get the students thinking through a situation. Also, GSP is used throughout the unit; sometimes as a worksheet in which students fill in answers while looking at problems, sometimes as a tool for experimenting and conjecturing, and other times for modeling and solving problems. Additionally, most lessons start with a problem that leads into the lesson so that students see how the need for more sophisticated ways of thinking arise from real life problems. (Note: Although the theorems are stated on the last page

4 of this document, the lessons will make more sense if you have a copy of the textbook to look at in front of you.) Lesson Summaries: Lesson 1 In this lesson, though the worksheet is interactive, students will use GSP merely to type in answers to questions which should lead them into discovering the general equation for a circle. By doing the work on GSP, students also become more familiar with it for future activities. It is not as inquiry-based as the other lesson, but this is because it is developing a somewhat difficult concept. Lesson 2 This lesson starts with a problem, involving satellites, in which students first gain a grasp of it though the use of manipulatives and measurement (circles, strings, protractors, etc.). They then move on to GSP to explore the situation though inquiry and discovery. The initial questions leads the students thorough all the theorems they must learn, and at the end, the students apply these theorems to find the exact answer to the original problem. Lesson 3 In this lesson, students again start with manipulatives (a Mira, compass, ruler, and can (for circle tracing)) to explore a problem and then move on to GSP to discover more theorems about chords and arcs. Lesson 4 This lesson begins with a problem that is not as clearly related to the real world as the others. An interest in problem is developed however because of the surprising results of inscribing a quadrilateral in a circle. Students again switch to GSP to do more investigating and they are eventually able to understand and prove the results of the question after discovering some new theorems. Lesson 5 This lesson s theorems are shown in a way that they build directly off of the ones from the previous lesson. Even so, the theorem that the students are trying to formalize isn t easy. Students begin the class using GSP this time. The students eventually use GSP s very unique graphing capabilities to provide another model for discovering the relationship between the different parts of the problem.

5 Lesson 1: Introduction to Circles Discovering the Equation of a Circle Name: Dennis Kapatos Grade: 11 Subject: Geometry Materials and Handouts: Teacher s Computer with GSP Projector Student s Computers with GSP GSP Worksheet File: Problem with Watering Crops Textbooks Graphing Calculators Lesson Objectives: Students will be able to write and apply the two general equation of circle, one centered at the origin and one at any point (h,k) (Theorem 12.1). Students will be able to state where the two general equations come from and derive them using this knowledge. Students will be able to graph a circle on their graphing calculators. Anticipatory Set: 1. Teacher will have review questions on the board including what the distance formula is and some examples to apply it to. 2. Teacher will go over the answers with the students where the distance formula comes from. Developmental Activity: 3. Teacher will talk about the farming industry in this country and how more food is needed from less land using less labor. 4. Students will log onto their computers and open the GSP worksheet file: Problem with Watering Crops and begin working on it by themselves. 5. Students will work until they finish question Teacher will go over and discuss the answers the students came up with. 7. Students will continue working until the finish question Again the teacher will go over and discuss the answer the students came up with paying special attention to question 14 (it can be confusing). 9. The students will complete the rest of the worksheet. 10. Again the teacher will go over and discuss the students answers 11. The students will submit the completed worksheet to the teacher using the computer. 12. The students will complete the reflection questions to be added to their notebooks. 13. Finally, the teacher will show the students how to use their graphing calculators to graph circles and discuss why a positive and negative form of the same equation are necessary to get both halves of the circle. Teacher will give an example of a system of equations to show how this is useful. Homework: On page 589 of the textbook, questions 1-18, 21-30, 32, 34, and 42

6 Problems with Watering Crops Name: Date: Answers in Boxes in Red Farmer Bob is using a new high-tech watering system which uses a computer and many high powered sprinklers to water his crops for him. To organize his land, Bob decides to divide his entire field into 100 meter square sections. Sprinklers are placed strategically in order to water as much of the land as possible without overlap (see figure). 1) What do you notice about this arrangement of sprinklers? Some areas don t get watered. The property line for Bob s Land is in Red. The 100 meter square sections are in blue. The red dots are the sprinkler heads with their spraying area shown in green. 2) Estimate: About what percent of Bob s fields do you think are not getting any water? Responses Vary.

7 Notice that some areas of Bob s field get no water at all. The computer uses the center sprinkler as the origin, that is, it assigns it the coordinates, (0,0). All other points on Bob s field, such as the point (100,0), are measured from this point. (0,0) (100,0) Notice that that this point, (90,60), isn t getting any water. 3) A water sensor placed in the ground tells Bob that the coordinates of one of the points which is not receiving water are (90,60). Fortunately, the sprinklers can be programmed to spray different distances. What distance should Bob tell the sprinkler at point (0,0) to spray in order for it to reach this point? Obviously you could set the sprinkler to spray a much larger area then needed, but his would waste water. What is the exact distance from the sprinkler to the dry point? (Show all work.) d = ( (90-0) 2 + (60-0) 2 ) 108 meters 4) There are other points in this area as well. What distance should Bob tell the sprinkler to spray in order to water any given point (x,y)? ( (x) 2 + (y) 2 ) Distance = 5) Bob s computer refers to distance as the letter r and it doesn t like square root signs either. How could you rewrite the equation above so Bob can enter the distance into the computer? r 2 = (x) 2 + (y) 2 = The above equation will give the sprinkler a new spraying radius that will reach any point (x,y) on the coordinate grid. It is the general equation of a circle with its center located at the point (0,0) and with a radius of r. 6) What is the radius and center of the circle formed by the equation 5 2 = x 2 + y 2? 7) What are they for the circle 36 = x 2 + y 2? r = 6, cntr = (0,0) r = 5, cntr = (0,0) 8) What is the equation of a circle with center (0,0) and radius 10? With radius 4? 100 = x 2 + y 2 16 = x 2 + y 2

8 9) What is the equation of the circle for the sprinkler at (0,0), before it s setting was changed? = x 2 + y 2 10) Name 4 points that lay on the edge of this circle. (100,0), (0,100), (-100,0), (0,-100) Bob decides that changing the range of all the sprinklers will create too much overlap and thus waste too much water (water is expensive in this part of the country), so he decides to install a new sprinkler at the point (100,58). (0,0) (100,0) 11) If this new sprinkler can exactly reach point (100,43) then what is its spraying radius? Use the distance formula and show your work. r = ( ( ) 2 + (58-43) 2 ) = 15 meters 12) If this new sprinkler can exactly reach point any point (x,y) then what is its spraying radius? (Note: This will be an equation in terms of r, x, and y.) ( (100 - x) 2 + (58- y) 2 ) r = r 2 = (100 - x) 2 + (58- y) 2 Now rewrite this without a square root sign: 13) Inside the pair of parentheses of this equation should be minus signs. If the x or y come after the minus sign then switch them with the other number. Write this equation. r 2 = (x - 100) 2 + (y - 58) 2 14) What is this the same as doing and why is it allowed in this case? That is, why doesn t it change the validity of the equation in this case? Convince yourself that this doesn t change anything before moving on. Reversing the order is the same as multiplying everything inside the parentheses by a -1, which, because the result is squared afterwards, doesn t effect the equation. (a - b) 2 = (-1 (a - b)) 2 = (-a + b) 2 = (b a) 2

9 15) Fill in the blank: The equation in #14 is the general equation of a circle with its center located at the point and with a radius of r (100,58) 16) Bob wants to install a lot of these new sprinklers in all the other dry areas as well. What is the equation for a circle that is centered at any given point (h,k) and that can reach any point (x,y)? (Again make sure that the x and y come first.) r 2 = (x - h) 2 + (y - k) 2 Now with this form of the equation, you can write the equation of any circle you could possibly imagine. 17) What is the radius and center of the circle 7 2 = (x-1) 2 + (y-1) 2? 18) What are they for the circle 4 = (x+3) 2 + (y-1) 2? 19) What is the equation of a circle with center (4,-5) and radius 9? r = 7, cntr = (1,1) r = 2, cntr = (-3,1) 81 = (x - 4) 2 + (y + 5) 2 20) What is the equation of a circle with center (-1,0) that passes through the point (3,2) 20 = (x + 1) 2 + y 2

10 Reviewing Major Ideas: Problem with Watering Crops Because this is a reflection it is for the students use only. Also responses will vary _ 1) Describe how the distance formula and the equation of a circle centered at the origin are related? Why does this intuitively make sense? 2) Will farmer Bob ever get this entire field watered without any overlapping areas? Why or why not? 3) Write the general equations that you found below for future reference. THM 12.1 General Equation of Circle with Radius r Centered at the Origin: THM 12.2 General Equation of Circle with Radius r and Center (h,k): 4) How did you determine the first equation? What did you do? 5 How did you determine the second equation? What did you do? 6) It s easy to make mistakes when first encountering applying these formulas. What mistakes did you make in questions 6-8 and (such as squaring r, making h negative, etc)? What things can you think of that will help you remember not to make them again?

11 Lesson 2: Tangents Discovering and Applying Tangents to Real Life Situations Materials and Handouts: Teacher s Computer with GSP Projector Student s Computers with GSP Textbooks Calculators String Measuring Tape Rulers Protractor Pre-measured Cardboard Circles Name: Dennis Kapatos Grade: 11 Subject: Geometry Lesson Objectives: Students will be able to write/describe and prove Theorems 12.2 through Students will be able to model problem situations using circles, tangents, radii, and other lines drawn to a circle. Given a geometric figure with missing measurements, students will be able to apply properties of circles, radii, and tangents to find them. Students will be able to work cooperatively to find an estimate to a given geometric problem. Reviewing Homework: 1. Answers to the homework questions will be shown with the projector. Students will correct their own homework. 2. Teacher will go over any problems the majority of the students had difficulty with. Developmental Activity: 3. Teacher will present today s problem situation involving satellites on white board. 4. Teacher will discuss with the class how they might solve this problem. They will decide to first do a rough scale model using a cardboard circles and string and take a more sophisticated approach later. 5. Teacher will split class into groups of 5. There will be a 2 people to hold the ends of the tangent strings (though they won t call it that, yet), 1 satellite person to hold the two strings together at the proper distance, 1 person to measure the central angle, and 1 person to record the data. 6. Each group will find their own measurement and the class will share their findings afterwards. 7. In order to find a more exact answer, Teacher will ask the class to break up their groups and go to their computers to draw the situation on GSP. 8. Teacher will talk the students through how to create the drawing using the teacher s instructions below. Teacher will ask questions when indicated.

12 9. Teacher will tell students to enter the 3 new theorems into their notes as they go, they ll prove 12.4 for homework. 10. After experimenting with GSP, teacher will work with student to solve original question (they will have to identify a right triangle and use some right angle trigonometry. 11. Teach will have student s summarize what they ve learned (the three theorems) and how they used the theorems to get an exact answer. Homework: Read proof of Theorem 12.2 on page 594, prove Theorem 12.4, and on page 596 of the textbook questions 1-8, 14-16, and 22

13 Satellites There are thousands of satellites circling the Earth right now tens of thousands of miles from the surface. They are used for any from relaying cell phone calls and televisions programming, to tracking weather systems and locating ships using GPS (global positioning systems). They can only see a portion of the earth s surface at a time but the higher a satellite s altitude, the more it can see. How far can the satellite see around the Earth (in degrees) from an altitude of 22,236 miles? (Note: The radius of the Earth is 3960 miles.) The above altitude puts the satellite at a geosynchronous orbit, how much could it see if it had a higher orbit? Satellite Earth Altitude = 2.72 cm Visible Area = 5.17 cm Can a satellite see half-way around the earth, that s 180 degrees, if it has a high enough orbit? Why?

14 Words in () are what the students should be guided to conjecture. They should not be told to them. Teacher s Instructions Instructions (say aloud): Construct a circle AB Construct line AC through the center of circle AB; hold down shift to make it vertical Construct point D on line AC Construct ray DE and ray DF as shown Hide points B and C and line AC Construct segment DA Construct point G, the intersection of segment DA and circle A Swing rays DE and DF till they exactly touch the circle, like you did with the string Construct radius AH and AI to the points where it looks like the rays touch the circle, make sure H and I are on the rays, not the circle Questions: Does this seem like a very accurate way of constructing this? What do you notice about the radii and rays? (they almost look perpendicular) Check this, measure angle AID and angle AHD, what did people get? Let s explore this further Instructions: Construct a new circle to the right Construct a radius Select it and its endpoint and construct a perpendicular line Questions: What happens when you move the point around? This line is called a tangent because it intersects the circle at exactly one point. That point is called the point of tangency. What should our definition of a tangent be? This is Theorem 12.2 and From any one point outside the circle, what how many different tangents can be drawn to one circle? Instructions: Construct a line through the center of the circle like you did for the other circle Double click it to mark it as a mirror line Select the point of tangency, the tangent, and the radius and reflect them Hide the mirror line and it s point Construct the intersection of these 2 tangents. Hide the 2 tangent lines Construct the tangent segments Construct the center segment and the point where it intersects the circle as shown Move the first point of tangency that you made around Questions: What do you notice about the lengths of the two tangent segments? (they re always the same length) Check this, measure them. Now move them around. does this conjecture check? This is Theorem Now getting back to our question, we know all we need to in order to solve this question. E B C D A m! A H D = G m! A ID = E H D A V I W M F F U O

15 Lesson 3: Properties of Chords and Arcs Using Properties of Chords and Arcs to Solve Problems Materials and Handouts: Teacher s Computer with GSP Projector Student s Computers with GSP GSP Worksheet Files Textbooks Graphing Calculators Compasses Rulers Mira s Empty Cans Name: Dennis Kapatos Grade: 11 Subject: Geometry Lesson Objectives: Students will be able to utilize Theorems 12.8 to construct the center of a circle. Students will be able to write/describe and prove Theorems 12.6, 12.5, 12.7, and Students will be able to model problem situations using circles, tangents, radii, and other lines drawn to a circle. Given a geometric figure with missing measurements, students will be able to apply the theorems learned so far to find them. Students will be able to work cooperatively to make and test conjectures of geometric figures. Reviewing Homework: 1. Answers to the homework questions 1-8, 14-16, and 22 will be shown with the projector. Students will correct their own homework. 2. Teacher will go over any problems the majority of the students had difficulty with. 3. Teacher will ask one student to present their proof of Theorem 12.4 to the class for discussion. Anticipatory Set: 4. Teacher will have students construct a point, a circle around this point, and then ask them to construct the circle s tangent using only a compass and ruler. 5. Teacher will ask students questions to remind them of relationship between tangent and radius to the point of tangency. Developmental Activity: 6. Teacher will ask student how they would construct a tangent without being given the center of the circle, only the edge. 7. Teacher will present today s problem situation involving a satellite dish. 8. Teach will have students work with a partner, each doing his/her own work but just sharing thoughts.

16 9. Teacher will tell students to construct an arc of a circle using the bottom of a can, or anything else that will serve this purpose. The idea is that they don t have the hole at the center like they would from using a compass. 10. Teacher will ask students how they could find the center if they used a Mira. If students need help, teacher will tell them to construct a point near both ends of the arc. 11. If students need more help, teacher will tell them to use their Mira find a position where it maps one of these points onto the other. 12. If students still need more help, tell them to draw this line that the Mira is on top of when it maps the two points onto each other. 13. Teacher will ask the students questions about what this line is and lead them to the idea that this line must go through the center because it is a line of symmetry for the circle. 14. Students should recognize the need to repeat this to obtain another line and an intersection. 15. Teacher will ask students to connect the two pairs of points they mapped to each other with segments at tell them that they are called chords. 16. Teacher will ask students to come up with a definition for a chord. 17. Teacher will ask what they constructed to the chord that found the center (a perpendicular bisector to the chord, be definition of a perpendicular bisector). 18. Teacher will have students make conjecture of this (Theorem 12.8). 19. Teach will have student break groups, go to their computers, and start up GSP 20. Teacher will have student s open a pre-made sketch of a circle, a chord, and its perpendicular bisector. 21. Student s will more around the points to see that any chord s perpendicular bisector crosses the center of the circle (see first picture). 22. Teacher will discuss with class and have students formalize their conjecture and write Theorem 12.8 in their notes. 22. Teacher will have students look on anther page of this sketch of the satellite from the beginning problem. 23. Teacher will ask students to find the placement of the receiver (see second picture). 24. Teacher will students look at other pages of the sketches which will have them discover Theorems 12.5, 12.6, and They are very obvious and thus not too much time is devoted to them. (see third and fourth pictures). 25. Students will enter these theorems into their notes. 26. Students will spend rest of class trying to solve question 20 on page 604.? Homework: Length D'E on m D'E = 4.83 cm BC = 5.13 cm E drag a a = Prove Theorem 12.9, and on page 603 of the textbook, questions 1-13, 18, and finish 20. G' D' B Length FG' on BC = m G'G = 4.83 cm F m AB = 3.27 cm G B C A m CD = 3.3 m GE = 3.09 cm E F D m EF = 3.06 cm

17 Locating the Center of the Satellite Dish A satellite dish in the shape of an arc (a portion of a circle) receives information by reflecting it off of a dish onto a receiver. During a strong hurricane, a piece of flying debris broke the receiver off. How could we find the exact place where to put a replacement receiver in order to receive a signal again??

20 Lesson 5: Angles Formed by Chords and Secants Coordinate Graphing with GSP Using Graphs to Determine Geometric Relationships Materials and Handouts: Teacher s Computer with GSP Projector Student s Computers with GSP GSP Worksheet Files Textbooks Graphing Calculators Name: Dennis Kapatos Grade: 11 Subject: Geometry Lesson Objectives: Students will be able to write/describe and prove Theorem and Students will be able to recognize real world applications of circles and the theorems learned so far. Students will be able to use coordinate graphs to determine the relationship between parts of geometric figure. Students will be able to work cooperatively to prove geometric theorems. Given a geometric figure with missing measurements, students will be able to apply the theorems learned so far to find them. Reviewing Homework: 1. Answers to the homework questions 1-12, 15, 16, 24-27, and 28 will be shown with the projector. Students will correct their own homework. 2. Teacher will ask each student to share the real life applications they have come up with using the projector to show their drawings etc.. Anticipatory Set: 3. Students will look at the sketches they made on GSP from lesson and teacher will ask questions to help them remember what theorems they learned. Developmental Activity: 4. Students will still be on GSP. 5. Teacher will talk the students through how to create the drawing using the teacher s instructions below. Teacher will ask questions when indicated. Theorems will be conjectured and entered into notes as the discussions progress. 6. At the end of the teacher s instructions, the students will work in pairs to try to prove Theorem Teacher will help them though they shouldn t have too much trouble (they ve done harder proofs before). 7. This proof will lead nicely into Theorem which the students will also enter into their notes.

21 8. For rest of period, students will practice applying these new theorems to some of the homework questions. Homework: Questions 1-9, and on page 617 of the textbook.

22 Words in () are what the students should be guided to conjecture. They should not be told to them. Teacher s Instructions Instructions (say aloud): Construct a circle AB Hide point B Construct Segments CD and ED as shown Questions: These are a new type of line we haven t seen yet. They re called secants. How might we define them? m CE on AB = Instructions: Construct the intersections, F and G, of these secants Measure arc angles CE and FG Measure angle D Move points C and/or E so that the measure of arc angle CE is a round number, say 60 degrees Questions: As you move point D onto the circle, how does this situation look familiar? (It is the inscribed angle theorem from last class.) What about when D is inside the circle? Outside? Lets explore the relationships between these angles. Instructions: Select the measurement of angle D then the arc angle of FG From the graph menu, choose plot as (x,y) Position your axis and change your unit values to make the graph fit nicely as shown The point that was created by the plot as (x,y) above, choose it and from the display menu, choose trace point Move point D around outside the circle C E A m FG on AB = F G D m! C D E =

23 m CE on AB = m FG on AB = C A F G D m! C D E = J E Questions: What is this plotting? What s going on? What is the relationship between angle D and the two arc angles you measured? (Students should work until they come up with the idea that the measure of angle A is half the difference of the two arc angles. They can get this through asking questions about the graph, its intercepts, it s slope, etc.) Instructions: Move points C and/or E so that they are both tangents Move D around outside the circle again Now move points C and/or E so that one is a secant and one is a tangent Move D around outside the circle a third time Questions: Does it matter weather DF and DG are both secants or tangents or combinations of the two? (No, the formula holds, it doesn t matter.) This is all three parts of Theorem 12.13, enter this into your notes, well prove it next Instructions: Hide all the measurements, axis, and gridlines Construct segments CG and FE as shown Make sure no lines are touching A Questions: (Teacher will have students attempt a proof (they know all they need to from lesson 4). This proof will also lead very nicely into Theorem 12.12, which students will enter into their notes.) E C A F G D

24 Theorems for Chapter 12: 12.1 The standard form of an equation of a circle with center (h,k) and radius r is (xh) 2 + (y-k) 2 = r If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency If a line in the same plane as a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle Two segments tangent to a circle from a point outside the circle are congruent In the same circle or in congruent circles, 1 congruent central angles have congruent arcs and, 2 congruent arcs have congruent central angels In the same circle or in congruent circles, 1 congruent chords have congruent arcs and, 2 congruent arcs have congruent chords A diameter that is perpendicular to a chord bisects the chord and its arc The perpendicular bisector of a chord contains the center of the circle In the same circle or in congruent circles, 1 chords equidistant form the center are congruent and, 2 congruent chords are equidistant from the center The measure of an inscribed angle is half the measure of its intercepted arc. Corollary 1 Corollary 2 Two inscribed angels that intercept the same arc are congruent. An angle inscribed in a semicircle is a right angle.

25 Corollary 3 The opposite angels of a quadrilateral inscribed in a circle are supplementary The measure of an angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc The measure of an angle formed by two chords that intersect inside a circle is half the sum of the measures of the intercepted arcs The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from an point outside the circle is half the difference of the measures of the intercepted arcs.

### 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

E XPLORING QUADRILATERALS E 1 Geometry State Goal 9: Use geometric methods to analyze, categorize and draw conclusions about points, lines, planes and space. Statement of Purpose: The activities in this

### 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

### 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

### 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

### 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)

### 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,

### 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.

### Mathematics Geometry Unit 1 (SAMPLE)

Review the Geometry sample year-long scope and sequence associated with this unit plan. Mathematics Possible time frame: Unit 1: Introduction to Geometric Concepts, Construction, and Proof 14 days This

### 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

### 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.

### 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

### 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

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

### Kristen Kachurek. Circumference, Perimeter, and Area Grades 7-10 5 Day lesson plan. Technology and Manipulatives used:

Kristen Kachurek Circumference, Perimeter, and Area Grades 7-10 5 Day lesson plan Technology and Manipulatives used: TI-83 Plus calculator Area Form application (for TI-83 Plus calculator) Login application

### 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

### 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

### Discovering Math: Exploring Geometry Teacher s Guide

Teacher s Guide Grade Level: 6 8 Curriculum Focus: Mathematics Lesson Duration: Three class periods Program Description Discovering Math: Exploring Geometry From methods of geometric construction and threedimensional

### Geometer s Sketchpad. Discovering the incenter of a triangle

Geometer s Sketchpad Discovering the incenter of a triangle Name: Date: 1.) Open Geometer s Sketchpad (GSP 4.02) by double clicking the icon in the Start menu. The icon looks like this: 2.) Once the program

### 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

### 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,

### 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 1. Unit 3: Perpendicular and Parallel Lines

Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples

### 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,

### 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

### Investigating Relationships of Area and Perimeter in Similar Polygons

Investigating Relationships of Area and Perimeter in Similar Polygons Lesson Summary: This lesson investigates the relationships between the area and perimeter of similar polygons using geometry software.

### 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

### 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

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

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

### 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

### 2.1. Inductive Reasoning EXAMPLE A

CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers

### Centers of Triangles Learning Task. Unit 3

Centers of Triangles Learning Task Unit 3 Course Mathematics I: Algebra, Geometry, Statistics Overview This task provides a guided discovery and investigation of the points of concurrency in triangles.

### 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

### 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.

### The Geometry of Piles of Salt Thinking Deeply About Simple Things

The Geometry of Piles of Salt Thinking Deeply About Simple Things PCMI SSTP Tuesday, July 15 th, 2008 By Troy Jones Willowcreek Middle School Important Terms (the word line may be replaced by the word

### 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

### Performance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will

Performance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will discover and prove the relationship between the triangles

### Unit 6 Trigonometric Identities, Equations, and Applications

Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean

### 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

### 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

### Tessellating with Regular Polygons

Tessellating with Regular Polygons You ve probably seen a floor tiled with square tiles. Squares make good tiles because they can cover a surface without any gaps or overlapping. This kind of tiling is

### 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

### Unit 2 - Triangles. Equilateral Triangles

Equilateral Triangles Unit 2 - Triangles Equilateral Triangles Overview: Objective: In this activity participants discover properties of equilateral triangles using properties of symmetry. TExES Mathematics

### 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

### 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

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

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

### Geometry 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

### MATHEMATICS Grade 12 EUCLIDEAN GEOMETRY: CIRCLES 02 JULY 2014

EUCLIDEAN GEOMETRY: CIRCLES 02 JULY 2014 Checklist Make sure you learn proofs of the following theorems: The line drawn from the centre of a circle perpendicular to a chord bisects the chord The angle

### 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

### 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.

### 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

### Linear Equations. 5- Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber

Linear Equations 5- Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber Tools: Geometer s Sketchpad Software Overhead projector with TI- 83

### 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

### The Use of Dynamic Geometry Software in the Teaching and Learning of Geometry through Transformations

The Use of Dynamic Geometry Software in the Teaching and Learning of Geometry through Transformations Dynamic geometry technology should be used to maximize student learning in geometry. Such technology

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

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

### Chapter 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.

### Heron s Formula. Key Words: Triangle, area, Heron s formula, angle bisectors, incenter

Heron s Formula Lesson Summary: Students will investigate the Heron s formula for finding the area of a triangle. The lab has students find the area using three different methods: Heron s, the basic formula,

### 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

### Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

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

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

### 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

### 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

### Tutorial 1: The Freehand Tools

UNC Charlotte Tutorial 1: The Freehand Tools In this tutorial you ll learn how to draw and construct geometric figures using Sketchpad s freehand construction tools. You ll also learn how to undo your

### Geometry Module 4 Unit 2 Practice Exam

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

### The Area of a Triangle Using Its Semi-perimeter and the Radius of the In-circle: An Algebraic and Geometric Approach

The Area of a Triangle Using Its Semi-perimeter and the Radius of the In-circle: An Algebraic and Geometric Approach Lesson Summary: This lesson is for more advanced geometry students. In this lesson,

### Precalculus. What s My Locus? ID: 8255

What s My Locus? ID: 855 By Lewis Lum Time required 45 minutes Activity Overview In this activity, students will eplore the focus/directri and reflection properties of parabolas. They are led to conjecture

### Lesson #13 Congruence, Symmetry and Transformations: Translations, Reflections, and Rotations

Math Buddies -Grade 4 13-1 Lesson #13 Congruence, Symmetry and Transformations: Translations, Reflections, and Rotations Goal: Identify congruent and noncongruent figures Recognize the congruence of plane

### 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,

### Chapter 18 Symmetry. Symmetry of Shapes in a Plane 18.1. then unfold

Chapter 18 Symmetry Symmetry is of interest in many areas, for example, art, design in general, and even the study of molecules. This chapter begins with a look at two types of symmetry of two-dimensional

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

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

### A Correlation of Pearson Texas Geometry Digital, 2015

A Correlation of Pearson Texas Geometry Digital, 2015 To the Texas Essential Knowledge and Skills (TEKS) for Geometry, High School, and the Texas English Language Proficiency Standards (ELPS) Correlations

### 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

### 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

### Lesson 26: Reflection & Mirror Diagrams

Lesson 26: Reflection & Mirror Diagrams The Law of Reflection There is nothing really mysterious about reflection, but some people try to make it more difficult than it really is. All EMR will reflect

### Circles, Angles, and Arcs

Here are four versions of the same activity, designed for students with different familiarity with Sketchpad and with different needs for specific support in the course of doing the activity. The activities

### 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

### Radius, Diameter, Circumference, π, Geometer s Sketchpad, and You! T. Scott Edge

TMME,Vol.1, no.1,p.9 Radius, Diameter, Circumference, π, Geometer s Sketchpad, and You! T. Scott Edge Introduction I truly believe learning mathematics can be a fun experience for children of all ages.

### 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:

### 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

### Circles in Triangles. This problem gives you the chance to: use algebra to explore a geometric situation

Circles in Triangles This problem gives you the chance to: use algebra to explore a geometric situation A This diagram shows a circle that just touches the sides of a right triangle whose sides are 3 units,

### 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

### TIgeometry.com. Geometry. Angle Bisectors in a Triangle

Angle Bisectors in a Triangle ID: 8892 Time required 40 minutes Topic: Triangles and Their Centers Use inductive reasoning to postulate a relationship between an angle bisector and the arms of the angle.

### Reflection and Refraction

Equipment Reflection and Refraction Acrylic block set, plane-concave-convex universal mirror, cork board, cork board stand, pins, flashlight, protractor, ruler, mirror worksheet, rectangular block worksheet,

### 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,

### Show all work for credit. Attach paper as needed to keep work neat & organized.

Geometry Semester 1 Review Part 2 Name Show all work for credit. Attach paper as needed to keep work neat & organized. Determine the reflectional (# of lines and draw them in) and rotational symmetry (order

### Math 531, Exam 1 Information.

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

### 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.

### D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

### 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,

### 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

### 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

### Teaching Guidelines. Knowledge and Skills: Can specify defining characteristics of common polygons

CIRCLE FOLDING Teaching Guidelines Subject: Mathematics Topics: Geometry (Circles, Polygons) Grades: 4-6 Concepts: Property Diameter Radius Chord Perimeter Area Knowledge and Skills: Can specify defining

### 39 Symmetry of Plane Figures

39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that

### CIRCLE COORDINATE GEOMETRY

CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### 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

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### 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