Arcs, Central and Inscribed Angles Coming Full Circle

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1 rcs, Central and Inscribed ngles SUGGESTED LERNING STRTEGIES: Shared Reading, Summarize/Paraphrase/Retell, Visualization Chris loves to play soccer. When he was seven years old, his family enrolled him in a soccer camp so that he could begin to learn the skills required to be a good player. For two years, Chris made progress with his team, but he wanted to improve. So this year, for Chris s ninth birthday, Mr. Green, a family friend who is an experienced soccer player, volunteered to coach Chris for a few hours every weekend. During their first sessions together, Chris handled the ball well as he ran up and down the field, but he had great difficulty kicking an accurate goal. Mr. Green suggested that they focus on improving this particular skill in future sessions. t every practice, Mr. Green stands centered between the goalposts and has Chris try to kick a goal from different positions on the field. He has noticed that Chris s shots go anywhere from right on target to 15 on either side of his target. With this information, Mr. Green believes that he and Chris can find locations on the soccer field from which Chris can be sure of kicking between the goalposts. lthough his shots may then be blocked by the goalie, at least Chris will stand a better chance of making a goal. Your teacher will provide you with a diagram of part of the soccer field, the 24-foot-wide goal, and the point X at which Mr. Green plans to stand. Points and represent the goalposts, which are 24 feet apart. Chris will aim his kicks directly at Mr. Green from various points on the playing field to try to find the locations where, even with his margin of error, his shots will land between the goalposts. Your teacher will also provide you with a diagram of a 30 -angle that has a bisecting ray. The vertex S represents the point from which Chris makes his kick, the bisecting ray represents the path to the target at which Chris is aiming, and the sides of the 30 -angle form the outer boundaries of Chris s possible kicks, given that his margin of error is up to 15 on either side of the target. CTIVITY 4.2 Unit 4 Circles and Constructions 285

2 CTIVITY 4.2 rcs, Central and Inscribed ngles SUGGESTED LERNING STRTEGIES: Close Reading, Think/ Pair/Share, Use Manipulatives, Questioning the Text, Visualization, Create Representations, Look for a Pattern Your teacher will give you sheets containing the soccer field diagram and the 30 -angle diagram. The angle diagram will be used as a tool in estimating the outer boundaries of Chris s kicks when he aims at point X from various locations on the soccer field. 1. On the soccer field diagram, you will notice three points labeled Points 1, 2, and 3. These three points represent the different positions on the field from which Chris will attempt his shot at the goal. Place the vertex of your angle at Point 1 and make certain that point X lies on the angle bisector. Will Chris s shot be guaranteed to end up between the goalposts from this position on the field? Explain. 2. One at a time, place the vertex of the angle on Point 2 and then on Point 3. Each time, make certain that point X lies on the angle bisector. Determine whether Chris s shots are guaranteed to end up between the goalposts from these positions on the field. Which, if either, position, is a sure shot at the goal for Chris? 3. With experimentation, you should find that there is a region of the playing field from which Chris is certain to have a shot into the goal zone, despite his margin of error. Use the soccer field diagram and the angle diagram to test points on the field until you can make an informed conjecture as to the shape of this region. Write a description of the region. On the soccer field diagram, clearly identify at least eight points on the outer boundary of this region. 286 Springoard Mathematics with Meaning TM Geometry

3 rcs, Central and Inscribed ngles CTIVITY 4.2 SUGGESTED LERNING STRTEGIES: ctivating Prior Knowledge, Close Reading, Interactive Word Wall, Vocabulary Organizer 4. elow is a diagram of circle O. O and O are called. O Q CDEMIC VOCULRY central angle Points and divide the circle into two arcs. The smaller arc is known as the minor arc, and the larger arc is known as major arc Q. The angle formed by the two radii, O, is called a central angle of this circle. In general, a central angle is an angle whose vertex is at the center of a circle and whose sides contain radii of the circle. n arc intercepted by a central angle is the minor arc that lies in the interior of the angle. Notice that the major arc associated with points and lies outside O, while the minor arc lies in the interior of O. is said to be intercepted by O. y definition, the measure of a minor arc is equal to the measure of the central angle that intercepts the minor arc. The notation for the measure of is m. The notation for a minor arc requires the endpoints of the arc,. The notation for a major arc requires a point on the arc included between the endpoints of the arc, Q. Semicircles are named as major arcs. The measure of a minor arc must be between 0 and 180. The measure of a major arc must be at least 180 and less than 360. The measure of a semicircle is 180. Unit 4 Circles and Constructions 287

4 CTIVITY 4.2 rcs, Central and Inscribed ngles SUGGESTED LERNING STRTEGIES: Think/Pair/Share, Quickwrite TRY THESE Given the circle below with center C, diameters JR and KQ and m RCQ = 50. Use the definitions for central angle and intercepted arc along with triangle properties to find each of the following. a. m RQ = b. m JQ = c. m CRQ = d. m JQR = e. m JKQ = f. JK. Write a definition for congruent arcs. J K C Q R TRY THESE Given the circle below with center C and diameters JR, KQ and PL. PL JR and KQ bisects PCR. P Q a. Explain why PQ QR. J C 2 1 R b. m 1 + m 2 = c. m JP + m PQ = m. Explain. d. m JQ = e. m JRQ = f. m JL = g. m JRL =. Explain. K L 288 Springoard Mathematics with Meaning TM Geometry

5 rcs, Central and Inscribed ngles CTIVITY 4.2 SUGGESTED LERNING STRTEGIES: Use Manipulatives, Interactive Word Wall, Vocabulary Organizer, Think/Pair/ Share, Create Representations, Quickwrite, Questioning the Text O 5. Use a protractor to find the measure of central angle O. m O = and m = 6. Choose any point on the major arc and label the point P. Draw P and P. P and P form an inscribed angle. a. Name the arc intercepted by P. CDEMIC VOCULRY inscribed angle b. List the characteristics of an inscribed angle. 7. Use a protractor to find the measure of inscribed angle P. m P =. 8. Draw a different point R on the circle. Then draw a new inscribed angle that has a vertex R and that intercepts. Find the measure of the new inscribed angle. m R =. 9. Make a conjecture about the measure of any inscribed angle of this circle that intercepts. Test your conjecture by creating and measuring three more inscribed angles that intercept. Unit 4 Circles and Constructions 289

6 CTIVITY 4.2 rcs, Central and Inscribed ngles SUGGESTED LERNING STRTEGIES: Think/Pair/Share, Look for a Pattern, Quickwrite 10. Refer to Try These, Try These, and Items 5 and 7 to complete the following table. Try These Try These Try These Items 5 and 7 Measure of the intercepted arc Measure of the central angle Measure of the inscribed angle m JQ = m JCQ = m JRQ = m JQ = m JCQ = m JRQ = m JL = m JCL = m JRL = m = m O = m P = 11. ased upon any patterns that you see in the table above, write a conjecture about the relationship between the measure of an inscribed angle and the measure of the central angle that intercepts the same arc. The Proof The proof for the conjecture that you wrote in Item 11 examines each of the three possible positions of the center of the circle relative to the inscribed angle: (Case 1) the center lies on a side of the inscribed angle; (Case 2) the center is in the interior of the inscribed angle; and (Case 3) the center lies outside of the inscribed angle. If we can prove the conjecture to be true for each of these three cases, we can form one theorem that covers all possible locations of the center of the circle relative to the inscribed angle. Case 1 Case 2 Case Springoard Mathematics with Meaning TM Geometry

7 rcs, Central and Inscribed ngles CTIVITY 4.2 SUGGESTED LERNING STRTEGIES: Group Presentation, Think/Pair/Share, Identify a Subtask, Quickwrite 12. The figure below shows a circle with center O. The values a, b, c and d refer to the degree measure of each indicated angle. Identify each of the following. O a d b C c a. the degree measure of the central angle intercepting b. the degree measure of the inscribed angle intercepting c. ccording to your conjecture in Item 11, b = Use the circle above and triangle properties to complete each of the following key steps in a proof for Case 1 of your conjecture. a. ecause O is an exterior angle to OC, a =. b. O. Explain. c. OC is an triangle d. b =. Explain. e. Use your responses for Parts a and d to verify your response to Part c in Item 12. Unit 4 Circles and Constructions 291

8 CTIVITY 4.2 rcs, Central and Inscribed ngles SUGGESTED LERNING STRTEGIES: ctivating Prior Knowledge, Group Presentation, Think/Pair/Share, Identify a Subtask, Quickwrite 14. The figure below shows a circle with center O, which (according to Case 2 for the proof of your conjecture in Item 11) is in the interior of the inscribed angle, P. For the angles shown below, use the conjecture you made in Item 11 to complete the following equation. m P = 1 m. 2 P a O c d e f C b 15. The diameter CP has been drawn as a dotted segment. Use the lower case variables in the circle above and the results from Items 12 and 13 (Case 1) to complete the following: a. c = 1 1 and d = 2 2 b. m O = + and m P = +. Explain. c. Write a complete and convincing argument for Case 2 that explains the statement that you completed in Item Springoard Mathematics with Meaning TM Geometry

9 rcs, Central and Inscribed ngles CTIVITY 4.2 SUGGESTED LERNING STRTEGIES: Group Presentation, Think/Pair/Share, Identify a Subtask, Quickwrite 16. The figure below shows a circle with center O, which (according to Case 3 for the proof of your conjecture in Item 11) is in the exterior of the inscribed angle, P. Notice that once again the diameter PC has been drawn as a dotted line and the lower case variables represent the degree measures of the indicated angles. b c P a d O C Use the lower case variables in the diagram above to complete each of the following. a. c = 1 1 and b + c = 2 2 b. Write a complete and convincing argument for Case 3 that explains why the measure of the inscribed angle that intercepts is half the measure of the central angle that intercepts. Unit 4 Circles and Constructions 293

10 CTIVITY 4.2 rcs, Central and Inscribed ngles SUGGESTED LERNING STRTEGIES: Quickwrite, Self/Peer Revision, Group Presentation, Think/Pair/Share 17. We have now proven the conjecture from Item 11 for each of the three possible locations of the center of the circle relative to the inscribed angle. Since the conjecture has been proven to be true for each, we can now formulate one theorem that includes all three cases. Complete the Inscribed ngle Measure Theorem below: Inscribed ngle Measure Theorem In a circle, the measure of an inscribed angle is one-half. TRY THESE C Given the circle with center, C, below, and m HC = 50. Find each of the following. H a. m HI = b. m HCR = c. m HIR = d. m IR = R C I P 18. Given the circle to the right with PL US. a. Complete the following proof: Statements 1. PL US Reasons 1. Given If two parallel lines are cut by a transversal, alternate interior angles are. 3. m 1 = m 2 = Substitution PS 6. L U 1 2 S b. Write the theorem suggested by this proof. 294 Springoard Mathematics with Meaning TM Geometry

11 rcs, Central and Inscribed ngles CTIVITY 4.2 SUGGESTED LERNING STRTEGIES: Think/Pair/Share, Use Manipulatives t the beginning of this activity, Chris and Coach Green were trying to determine the locations on a soccer field from which Chris could kick a goal and be confident that by aiming at the center of the goal zone he could place the ball between the goalposts. In Item 3, you and your group made a conjecture about the region of the field that is determined by all of the points from where Chris is certain to make a goal. Let s see how the Inscribed ngle Measure Theorem we proved might either support your conjecture or help you to revise it. In the circle below, the inscribed angle is P and m P = 30. PX bisects P. X P 19. Place the 30 -angle diagram that you used at the beginning of the activity so that the vertex, S, is at point P and so the angle bisectors coincide. Slide S so that it is closer to X than P, keeping the angle bisectors on top of each other. Then slide S away from X, so that S is outside the circle and so that angle bisectors are still aligned. Think about locations of S from which Chris will be certain to make a shot into the goal zone. Using the circle as a point of reference, from which points along PX will Chris be certain of making a shot into the goal zone? Unit 4 Circles and Constructions 295

12 CTIVITY 4.2 rcs, Central and Inscribed ngles SUGGESTED LERNING STRTEGIES: Think/Pair/Share, Create Representations, Quickwrite, Group Presentation, Use Manipulatives, Self/Peer Revision 20. On the circle below, select a new point W on the major arc determined by points and, and carefully draw W. X 21. m W =. Explain. 22. y careful use of a protractor or by construction, draw the angle bisector of W. Does your new angle bisector also go through point X? Use the properties that you have learned in this ctivity to support your conclusion. 23. Place vertex S of your 30 angle on the circle above so the angle bisectors coincide. Slide S from a starting position at W closer and farther from point X. Using the circle as a point of reference, from which points along this angle bisector will Chris be certain of making a shot into the goal zone? 24. Restate or revise your conjecture in Item 3 about the region in which Chris wants to be when kicking a goal. 296 Springoard Mathematics with Meaning TM Geometry

13 rcs, Central and Inscribed ngles CTIVITY 4.2 CHECK YOUR UNDERSTNDING Write your answers on notebook paper. Show your work. 1. The circle has center O. Let E IS and m TOE = 36. Find each of the following. a. m TE b. m ST c. m SIT d. m S e. m IS f. m IOT g. m IT h. m TI 2. Find the value of x in each of the following. a. 96 8x S O I T E 3. Use the circle below to complete the following. a. If C then. b. Write a convincing argument that supports your response to Part a. Hint: Draw C or the three radii that contain points, and C. 4. If a polygon is inscribed in a circle, then each of its vertices lie on the circle. Which of the following correctly depict an inscribed polygon? a. b. C b. c. 10x 5 16x m 1 = 7x + 4 and m 2 = 5x + 10 c. d. 5. Find the length of one side of a square inscribed in a circle that has a radius of 4 cm. a. 4 cm b. 4 2 cm c. 8 cm d. 8 2 cm Unit 4 Circles and Constructions 297

14 CTIVITY 4.2 rcs, Central and Inscribed ngles CHECK YOUR UNDERSTNDING () Write 6. Use your sometimes, answers always, on notebook or never to paper. make Show each your work. 8. a. m E =. Explain. of the following statements true. b. m F =. Explain. a. Triangles can be inscribed in a circle. c. Complete the table below. b. Trapezoids can be inscribed in a circle. m 1 m 2 x y c. Parallelograms that are not rectangles 40 can be inscribed in a circle. 90 d. Equilateral triangles can be inscribed in a semicircle. e. Right triangles can be inscribed in a semicircle. f. Trapezoids can be inscribed in a semicircle. 7. MTHEMTICL REFLECTION that can be inscribed in a circle? What must be true about the angles of any quadrilateral Use this figure and given information to answer Item 8. Given the circle with center C and diameter. DE is tangent to the circle at point. D C x y 2 1 F E a d. In each case, how does m 1 compare to y? e. Use the information in the table to complete the following. Theorem: The measure of an angle formed by a tangent to a circle and a chord that contains the point of tangency equals. 298 Springoard Mathematics with Meaning TM Geometry