Circles and Constructions

Transcription

1 Circles and Constructions Unit 4 Unit Overview In this unit you will study many geometric concepts related to circles including angles, arcs and segments. You will learn to write the equations of circles and spheres. You will also study geometric constructions.? Essential Questions Why is it important to understand geometric constructions? cademic Vocabulary dd these words to the academic vocabulary portion of your notebook. arc central angle chord circle construction inscribed angle sphere tangent? How are the geometric properties of circles, their angles, and their arcs used to model and describe real world phenomena? EMBEDDED SSESSMENTS This unit has two Embedded ssessments. These embedded assessments allow you to demonstrate your knowledge of the measures of angles, arcs and segments in circles as well as your ability to find the area of circles and write their equations. Embedded ssessment 1 ngles and Segments in Circles p. 319 Embedded ssessment 2 rea and rc Length, Equation of a Circle p

2 UNIT 4 Getting Ready Write your answers on notebook paper. 5. Find the measure of. 1. Simplify. a. 75 b Solve the following for x. a. x 2-7x + 10 = 0 b. 4(x + 2) = 6x c. x(x + 1) = 2(x + 1) 3. Find the following products a. (5 x 2 )(3x 2 ) b. 2y(3y + 5) 4. Find the distance between the following: a. (3, 4) and (-2, 6) b. (1, 3) and (5, 3) 6. In the figure below, C = 14 units, BD = 12 units and D = 20 units. Find the lengths BC and B. B 7. Compare and contrast the characteristics of a radius, diameter and chord of a circle. 8. Sketch a line, b, that is the perpendicular bisector of a segment C. Describe the geometric characteristics of your sketch. C D 276 SpringBoard Mathematics with Meaning TM Geometry

3 Tangents and Chords Off On a Tangent SUGGESTED LERNING STRTEGIES: Group Presentation, Think/Pair/Share, Quickwrite, Interactive Word Wall, Vocabulary Organizer, Create Representations, Quickwrite CTIVITY 4.1 circle is the set of all points in a plane at a given distance from a given point in the plane. Lines and segments that intersect the circle have special names. The following illustrate tangent lines to a circle. CDEMIC VOCBULRY circle CDEMIC VOCBULRY tangent is a line in the plane of a circle that intersects the circle at just one point, called the point of tangency. 1. On the circle below, draw three unique examples of lines or segments that are not tangent to the circle. MTH TERMS secant is a line that intersects the circle in two points. 2. Write a description of tangent lines. 3. Using the circle below, a. Draw a tangent line and a radius to the point of tangency. b. Describe the relationship between the tangent line and the radius of the circle drawn to the point of tangency. MTH TERMS The radius of a circle is a segment, or length of a segment, from the center to any point on the circle Unit 4 Circles and Constructions 277

4 CTIVITY 4.1 Tangents and Chords Off On a Tangent MTH TERMS The diameter is a segment, or the length of a segment, that contains the center of a circle and two end points on the circle. SUGGESTED LERNING STRTEGIES: Create Representations, Use Manipulatives, Notetaking, Quickwrite Chuck Goodnight dug up part of a wooden wagon wheel. n authentic western wagon has two different sized wheels. The front wheels are 42 inches in diameter while the rear wheels are 52 inches in diameter. Chuck wants to use the part of the wheel that he found to calculate the diameter of the entire wheel, so that he can determine if he has found part of a front or rear wheel. scale drawing of Chuck s wagon wheel part is shown below. CDEMIC VOCBULRY chord is a segment whose endpoints are points on a circle. 4. Trace the outer edge of the portion of the wheel shown onto a piece of paper. CDEMIC VOCBULRY 5. Draw two chords on your arc. n arc is part of a circle consisting of two points on the circle and the unbroken part of the circle between the two points. 6. Using a ruler, draw a perpendicular bisector to each of the two chords and extend the bisectors until they intersect. The perpendicular bisectors of two chords in a circle intersect at the center of the circle. 7. Determine the diameter of the circle that will be formed. Explain how you arrived at your answer. 8. The scale factor for the drawing is 1:12. Determine which type of wheel can contain the part Chuck found. Justify your answer. 278 SpringBoard Mathematics with Meaning TM Geometry

5 Tangents and Chords Off On a Tangent CTIVITY 4.1 SUGGESTED LERNING STRTEGIES: Look for a Pattern, Use Manipulatives, Quickwrite 9. In the circle below: Draw a diameter. Draw a chord that is perpendicular to the diameter. a. Use a ruler to take measurements in this figure. What do you notice? b. Compare your answer with your neighbor s answer. What conjecture can you make based on your investigations of a diameter perpendicular to a chord? Unit 4 Circles and Constructions 279

6 CTIVITY 4.1 Tangents and Chords Off On a Tangent SUGGESTED LERNING STRTEGIES: Create Representations, Notetaking, Self/Peer Revision 10. For the theorem below, the statements for the proof have been scrambled. Your teacher will give you a sheet that lists these statements. Cut out each of the statements and rearrange them in logical order. D X R C Y B MTH TERMS If two segments are the same distance from a point, they are equidistant from it. Theorem: In a circle, two congruent chords are equidistant from the center of the circle. Given: B CD ; RX CD RY B Prove: RY RX Draw radii RB and RD. RY RX B CD ; RX CD ; RY B B = CD DXR and BYR are right triangles. RB RD 1 1 B = 2 2 CD BY DX DXR and BYR are right angles. BY = DX BY = 1 1 B; DX = 2 2 CD DXR BYR 280 SpringBoard Mathematics with Meaning TM Geometry

7 Tangents and Chords Off On a Tangent CTIVITY 4.1 SUGGESTED LERNING STRTEGIES: Create Representations, Self/Peer Revision 11. The reasons for the proof in Item 10 are scrambled below. Your teacher will give you a sheet that lists these reasons. Cut out each of the reasons and rearrange them so they match the appropriate statement in your proof. Through any two points there is exactly one line. Definition of right triangle Definition of congruent segments Definition of congruent segments Multiplication Property C.P.C.T.C. HL Theorem Given Definition of perpendicular lines ll radii of a circle are congruent. diameter perpendicular to a chord bisects the chord. Substitution Property Unit 4 Circles and Constructions 281

8 CTIVITY 4.1 Tangents and Chords Off On a Tangent SUGGESTED LERNING STRTEGIES: Quickwrite, Think/Pair/Share E F R B 12. Given EF B, explain how you know that EF and B are not equidistant from the center, R. 13. Michael said that if two chords are the same length but are in different circles that are not necessarily concentric circles, then they will not be the same distance from the center of the circle. Is he correct? If he is, give a justification. If not, give a counterexample. 282 SpringBoard Mathematics with Meaning TM Geometry

9 Tangents and Chords Off On a Tangent CTIVITY 4.1 SUGGESTED LERNING STRTEGIES: Think/Pair/Share, Create Representations, Self/Peer Revision Theorem: The tangent segments to a circle from a point outside the circle are congruent. 14. Use the theorem above to write the prove statement for the diagram below. Then, prove the theorem. Given: BD and DC are tangent to circle. Prove: B C D MTH TERMS tangent segment to a circle is part of a tangent line with one endpoint outside the circle and the second endpoint at a point of tangency to the circle. 15. In the diagram, RT = 12 cm, RH = 5 cm, and MT = 21 cm. Determine the length of RM. Explain how you arrived at your answer. T H R O D M Unit 4 Circles and Constructions 283

10 CTIVITY 4.1 Tangents and Chords Off On a Tangent CHECK YOUR UNDERSTNDING Write your answers on on notebook paper. paper. Show Show your work. 5. In the diagram below, H, D, and DH are your work. each tangent to circle Q. T = 9, H = 13, and D = 15. HD =?. 1. Draw a circle. Draw a line that is neither a secant line nor a tangent line to the circle. H T 2. In the diagram, TP is tangent to circle D. Determine m DTP. T Q D In the diagram, C is tangent to circle P, the radius of circle P is 8 cm and BC = 9 cm. C =?. P B P C D M 6. Suppose a chord of circle is 5 inches from the center and is 24 inches long. Find the length of the radius of the circle. 7. MTHEMTICL REFLECTION Explain how to prove the following conjecture: If a diameter is perpendicular to a chord, then the diameter bisects the chord. 4. In the diagram, NM and QN are tangent to circle P, the radius of circle P is 5 cm, and MN = 12 cm. QN =?. P M Q N 284 SpringBoard Mathematics with Meaning TM Geometry

11 rcs, Central and Inscribed ngles Coming Full Circle 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 B 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

12 CTIVITY 4.2 rcs, Central and Inscribed ngles Coming Full Circle 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 SpringBoard Mathematics with Meaning TM Geometry

13 rcs, Central and Inscribed ngles Coming Full Circle CTIVITY 4.2 SUGGESTED LERNING STRTEGIES: ctivating Prior Knowledge, Close Reading, Interactive Word Wall, Vocabulary Organizer 4. Below is a diagram of circle O. O and OB are called. O Q CDEMIC VOCBULRY central angle B Points and B divide the circle into two arcs. The smaller arc is known as the minor arc B, and the larger arc is known as major arc QB. The angle formed by the two radii, OB, 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 B lies outside OB, while the minor arc lies in the interior of OB. B is said to be intercepted by OB. By 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 B is m B. The notation for a minor arc requires the endpoints of the arc, B. The notation for a major arc requires a point on the arc included between the endpoints of the arc, QB. 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

14 CTIVITY 4.2 rcs, Central and Inscribed ngles Coming Full Circle 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 B 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 SpringBoard Mathematics with Meaning TM Geometry

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

16 CTIVITY 4.2 rcs, Central and Inscribed ngles Coming Full Circle SUGGESTED LERNING STRTEGIES: Think/Pair/Share, Look for a Pattern, Quickwrite 10. Refer to Try These, Try These B, and Items 5 and 7 to complete the following table. Try These Try These B Try These B 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 B = m OB = m PB = 11. Based 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 SpringBoard Mathematics with Meaning TM Geometry

17 rcs, Central and Inscribed ngles Coming Full Circle 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 B a. the degree measure of the central angle intercepting B b. the degree measure of the inscribed angle intercepting B 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. Because OB is an exterior angle to BOC, a =. b. O. Explain. c. BOC 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

18 CTIVITY 4.2 rcs, Central and Inscribed ngles Coming Full Circle 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, PB. For the angles shown below, use the conjecture you made in Item 11 to complete the following equation. m PB = 1 m. 2 P a O c d e f C b 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 OB = + and m PB = +. Explain. c. Write a complete and convincing argument for Case 2 that explains the statement that you completed in Item SpringBoard Mathematics with Meaning TM Geometry

19 rcs, Central and Inscribed ngles Coming Full Circle 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, PB. 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 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 B is half the measure of the central angle that intercepts B. Unit 4 Circles and Constructions 293

20 CTIVITY 4.2 rcs, Central and Inscribed ngles Coming Full Circle 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 SpringBoard Mathematics with Meaning TM Geometry

21 rcs, Central and Inscribed ngles Coming Full Circle 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 PB and m PB = 30. PX bisects PB. X B 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

22 CTIVITY 4.2 rcs, Central and Inscribed ngles Coming Full Circle 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 B, and carefully draw WB. X B 21. m WB =. Explain. 22. By careful use of a protractor or by construction, draw the angle bisector of WB. 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 SpringBoard Mathematics with Meaning TM Geometry

23 rcs, Central and Inscribed ngles Coming Full Circle CTIVITY 4.2 CHECK YOUR UNDERSTNDING Write your answers on notebook paper. Show your work. 1. The circle has center O. Let BE IS and m TOE = 36. Find each of the following. a. m TE b. m S ST T c. m SIT d. m SB B E O e. m BIS f. m IOT g. m IT I h. m TBI 2. Find the value of x in each of the following. a. 96 8x 3. Use the circle below to complete the following. B a. If B BC then B. b. Write a convincing argument that supports your response to Part a. Hint: Draw C or the three radii that contain points, B 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

24 CTIVITY 4.2 rcs, Central and Inscribed ngles Coming Full Circle CHECK YOUR UNDERSTNDING () Write 6. Use your sometimes, answers always, on notebook or never to paper. make Show each your work. 8. a. m BE =. Explain. of the following statements true. b. m FB =. 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 64 in a semicircle. 150 e. Right triangles can be inscribed in a semicircle. a f. Trapezoids can be d. In each case, how does m 1 compare to y? inscribed in a semicircle. e. Use the information in the table to complete the following. 7. MTHEMTICL REFLECTION What must be true about the angles of any quadrilateral that can be inscribed in a circle? Use this figure and given information to answer Item 8. Given the circle with center C and diameter B. DE is tangent to the circle at point B. D C B x y 2 1 F E Theorem: The measure of an angle formed by a tangent to a circle and a chord that contains the point of tangency equals. 298 SpringBoard Mathematics with Meaning TM Geometry

25 ngles Formed by Chords, Secants, and Tangents In The Spotlight SUGGESTED LERNING STRTEGIES: Shared Reading, Marking the Text, Questioning the Text, ctivating Prior Knowledge, Visualization, Group Presentation, Think/Pair/Share, Quickwrite, Self/Peer Revision CTIVITY 4.3 The MIU School of Design is constructing a circular reflection pool. The pool has tiles along the inside edges that were designed by local artists and art students. For the tiles to have the desired visual effect, they need to be illuminated by spotlights. Students have been submitting their suggestions for the placement of the light fixtures. The light fixtures that are to be used for this project illuminate objects that are within 30 of the center of the bulb. The diagram below represents an overhead view of a single light fixture If the light fixtures are placed at the center of the circular pool and aimed outward towards the edge of the pool, how many would be needed to illuminate the entire pool? Explain. 2. If the same light fixtures in Item 1 are placed halfway between the center and the pool edge and aiming outward, about how much of the pool wall do you think would be illuminated? Estimate the number of additional light fixtures that would be needed to illuminate the entire pool. HINT: You may wish to draw a diagram with two concentric circles: one that represents the outer edge of the pool, and one that represents the points that are halfway between the center and the edge of the pool. 3. If the light fixtures were placed on the pool edge and aimed towards the center, how many light fixtures would be needed to illuminate the entire pool? Explain. Unit 4 Circles and Constructions 299

26 CTIVITY 4.3 ngles Formed by Chords, Secants, and Tangents In The Spotlight SUGGESTED LERNING STRTEGIES: ctivating Prior Knowledge, Think/Pair/Share, Create Representations, Self/Peer Revision ngles Formed by Chords Maury is considering a design that involves attaching two light fixtures back-to-back and placing the pairs in various locations in the pool. 4. The figure below represents an overhead view of the pool and one of the light fixture pairs located at the center of the pool, C. Find the degree measure of each of the illuminated portions of the pool, B and PQ. P C B Q 5. s Maury moves the pair of spotlights (point L in the figure below) left or right of the center, he notices that the sizes of the illuminated portions of the pool change. s one of the arcs increases in measure, the other arc decreases in measure. Maury needs to know if there is a relationship between the measure of the vertical angles, x, and the measure of the two intercepted arcs, a and b. a. Draw P. b. Each of the angles, x, is an exterior angle to LP. Therefore, x = m + m. c. Use the Inscribed ngle Measure Theorem to find the measure of PB and PQ. In terms of a and b. a B x x L Q P b d. Use your responses in Parts b and c to find an expression for x in terms of a and b. 300 SpringBoard Mathematics with Meaning TM Geometry

27 ngles Formed by Chords, Secants, and Tangents In The Spotlight SUGGESTED LERNING STRTEGIES: Think/Pair/Share, Self/Peer Revision, Visualization, Quickwrite CTIVITY 4.3 TRY THESE Use the figure below and the relationship that you discovered in Item 5d to find each of the following. a. If a = 40 and b = 80 then m 1 = and m 2 =. b. If a = 40 and m 1 = 65 then b = and m GR + m NI = c. If m GR = 100, m IN = 160 and m RI = 80 then m 1 =. d. If a = 4x - 4, b = 100 and m 1 = 5x + 3, write an equation and solve for x. G a N 1 T 2 I R b 6. Maury decided that he liked the effect when the back-to-back light fixture pairs were placed off-center because some of the tiles would be lit up more brightly than others. In the figure below, point represents the location of a pair of the spotlights, and the arcs WX and YZ represent the parts of the pool edge illuminated by the spot lights. a. Recall that m WX = 60. If m WX = 100 then m YZ =. b. m WX + m YZ =. c. If m WX = 100, then which arc in the figure represents the part of the pool edge where the tiles are most brightly lit? Explain. W X Y Z Unit 4 Circles and Constructions 301

28 CTIVITY 4.3 ngles Formed by Chords, Secants, and Tangents In The Spotlight SUGGESTED LERNING STRTEGIES: Visualization, Think/ Pair/Share, Create Representations, Quickwrite 7. In his design for the reflection pool, Maury placed three of the back-to-back light fixture pairs as depicted in the figure below. Points, B, and C represent the location of each pair. Each pair of lights is equidistant from the center of the circle. ES IG DN E D C S B I N G a. Will the entire pool edge be illuminated in this design? Explain. b. Shade in the region of the pool that will be illuminated by more than one spotlight. c. If ES is one fourth of the circumference of the pool, then m DE =. Show the work that supports your response. d. What fraction of the pool edge will be in brighter light than the rest of the pool edge? 302 SpringBoard Mathematics with Meaning TM Geometry

29 ngles Formed by Chords, Secants, and Tangents In The Spotlight SUGGESTED LERNING STRTEGIES: Shared Reading, Questioning the Text, Summarize/Paraphrase/Retell, Quickwrite, ctivating Prior Knowledge, Think/Pair/Share CTIVITY 4.3 ngles Formed by Tangents lessa realized that she could illuminate the pool with fewer spotlights by placing the spotlights outside the pool and pointing them towards the center. Even though a larger portion of the pool edge can be illuminated this way, there is a disadvantage: part of the pool edge will be in a shadow. M P N Q 8. If point represents the light source, which part of the pool edge will be illuminated and which part will be in the shadow? 9. lessa places the spotlight as close to the pool as possible, while at the same time illuminating the largest possible part of the pool edge. In the figure below, point represents the light source and point C represents the center of the pool. a. CP and CQ are called. b. P and Q are called. c. m PC = m QC = d. Recall that m = 60º. Find m C. (Hint: consider the angles in Quadrilateral PCQ.) P Q C e. What fraction of the pool edge will be illuminated by the spot light and what fraction will be in a shadow? Unit 4 Circles and Constructions 303

30 CTIVITY 4.3 ngles Formed by Chords, Secants, and Tangents In The Spotlight SUGGESTED LERNING STRTEGIES: Create Representations, Think/Pair/Share, Self/Peer Revision, Group Presentation, Quickwrite 10. In the diagram below, point C represents the center of the circle. P and Q are tangent to the circle. If m = x, then find an expression for m PQ in terms of x. P C Q TRY THESE B In the figure below, x is the degree measure of an angle whose sides are tangent to the circle and a and b represent arc measures (in degrees). Use the relationship that you discovered in Item 10 to find each of the following. a. Find a and b if x = 45. b. Find x if b = 100. c. Find x if a = 270. d. Solve for y if x = 4y and b = 20y -12. x b a 11. In her design, lessa decided to use three spotlights (as in Item 9) evenly spaced around the reflection pool. Draw a sketch of the overhead view of lessa s design. What fraction of the pool edge is in the shadows of a spotlight? What fraction of the pool edge is illuminated by two or more of the spotlights? 304 SpringBoard Mathematics with Meaning TM Geometry

31 ngles Formed by Chords, Secants, and Tangents In The Spotlight CTIVITY 4.3 SUGGESTED LERNING STRTEGIES: ctivating Prior Knowledge, Think/Pair/Share, Create Representations ngles Formed by Secants Even though she did not use them in her design, lessa investigated two additional situations in which the spotlight is located outside the circle. c T P b Q d R a 12. a. In the figure above, T and R are called because they each intersect the circle in two points. b. The points P, T, R, and Q divide the circle into four arcs. Which of the arcs lie in the interior of and which lie in the exterior? c. Which of the arcs are intercepted by? d. If the variables a, b, c, and d represent the measures of each of the four arcs, then a + b + c + d =. 13. Inscribed angles are formed when RT is drawn. In terms of a, b, c, and d, m 3 = and m 4 =. CONNECT TO P In calculus, you will study how the tangent and secant lines relate to the concept of a derivative. P b Q c d 3 R 4 T a Unit 4 Circles and Constructions 305

32 CTIVITY 4.3 ngles Formed by Chords, Secants, and Tangents In The Spotlight SUGGESTED LERNING STRTEGIES: Group Presentation, Think/Pair/Share, Create Representations, Quickwrite, Self/Peer Revision, Identify a Subtask P b Q c d 3 R 4 T a 14. a. Let x represent the measure of. x + m 3 + m 4 = b. Substitute the expressions that you found for m 3 and m 4 (in Item 13) into the equation that you wrote in Item 14a. Simplify your new equation. c. Refer to the equation in Item 12d. Solve this equation for c + d. d. Use your responses in 14b and 14c to find an expression for x in terms of a and b. e. Complete the following theorem: Theorem The measure of an angle formed by two secants drawn to a circle from a point in the exterior of the circle is equal to. TRY THESE C Use the relationship that you discovered in Item 14 and the figure below to find each of the following. x a. Find x if a = 125 and b = 35. b b. Find a if x = 35 and b = 40. c. Find x if a = 160, c = 80, and d = 70. c d d. Write an equation and solve for t if a = 10t, b = 3t - 10, and x = 4t - 1. a 306 SpringBoard Mathematics with Meaning TM Geometry

33 ngles Formed by Chords, Secants, and Tangents In The Spotlight CTIVITY 4.3 SUGGESTED LERNING STRTEGIES: Think/Pair/Share, Create Representations, Group Presentation, Simplify the Problem, Quickwrite, Self/Peer Revision diagram of the last situation that lessa investigated is shown below. P is tangent to the circle, and R is a secant. P a x b Q c R 15. The variables a, b, and c represent arc measures in degrees, and x represents the degree measure of. Write a true equation that involves the sum of the measure of the three arcs whose endpoints are P, Q, and R. 16. a. Draw PR. b. Write a true equation that involves the sum of the three angles in PR. c. Two of the angles in PR are also inscribed angles in the circle. Find the measures of those angles in terms of a, b, and c. d. Substitute the expressions that you found in Part c into your equation in Part b. e. Use your responses to Part d and Item 15 to find a simplified expression for m. Show your work. f. Complete the following theorem: Theorem The measure of an angle formed by a secant and a tangent drawn to a circle from a point in the exterior of the circle is equal to. Unit 4 Circles and Constructions 307

34 CTIVITY 4.3 ngles Formed by Chords, Secants, and Tangents In The Spotlight CHECK YOUR UNDERSTNDING Write 1. Solve your for answers t. on notebook paper. Show your work. 5. In Item 10, you found that m = b. 3t + 2 8t In many textbooks, is treated like the 15t 14 angles in Item 14 and m 2. Determine m 2 = 2 1 (a - b). if m 1 = 34 Write a clear and convincing b argument that shows the two 1 expressions 2 for m are equivalent. 3. If a circle is tangent to each side of a polygon, then the circle is inscribed in the polygon (and the polygon is circumscribed about the circle). Which of the following correctly depict a circle inscribed in a polygon? a. b. 6. farmer woke up one morning to find crop circles in his wheat field as shown below. If m P = 16 and m CO = 96 determine each of the following. W M P I N E O C a c. d. 4. Given m P = m SVW = 45, m ST = 80 and m SW = 30 Determine each of the following. W a. m QT b. m QR S V R c. m RST T Q P a. m WE b. m WIE d. m MO e. m M c. m OC f. m CNO g. If C OM, then determine m OC. 7. MTHEMTICL REFLECTION shown below. Imagine point moving out to the left to increase C. s it moves, and points of tangency B and D will change. B and D are tangent to the circle with center C as a. s C increases, what is happening to m? b. s C increases, what is happening to BD? c. How small does have to be for BD to be a diameter? Explain your answer. B D C 308 SpringBoard Mathematics with Meaning TM Geometry

35 Segment Lengths In Circles High-Quality Product SUGGESTED LERNING STRTEGIES: Interactive Word Wall, Summarize/Paraphrase/Retell, Predict and Confirm, Create Representations Lee is a mechanic who enjoys restoring cars in his spare time. In order to find the correct size for a part that he needs, Lee realizes that he needs to know the radius of a rotor. However, he can only measure part of the rotor without damaging other parts. Is there a way for Lee to find the radius of the rotor while avoiding damage to other parts? Perhaps some of the relationships between segments of chords, tangents and secants in circles would prove helpful. 1. Let s begin by examining chord segments. a. In the circle below, draw two intersecting chords. Name the chords M and TH intersecting at point O. CTIVITY 4.4 b. Draw MT and H. c. Name pairs of congruent angles from the diagram. Explain how you know each pair of angles is congruent. d. How are MOT and HO related? Justify your answer. Unit 4 Circles and Constructions 309

36 CTIVITY 4.4 Segment Lengths In Circles High-Quality Product SUGGESTED LERNING STRTEGIES: Close Reading, Create Representations e. Make a conjecture about the relationship between MO TO Justify your answer. and HO O. f. Is it true that MO O = HO TO? Why or why not? The relationship you just explored can be stated formally as a theorem. Theorem: If two chords of a circle intersect, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord. EXMPLE 1 Find B. ccording to the theorem, E 8 x B 6 4 D C B BC = So, x 4 = 4x = x = TRY THESE Find the value of x and y in each of the circles below. a. b. 8 4 x 10 x 9 3 x c x y 310 SpringBoard Mathematics with Meaning TM Geometry

37 Segment Lengths In Circles High-Quality Product CTIVITY 4.4 SUGGESTED LERNING STRTEGIES: Close Reading, Interactive Word Wall, Marking the Text There are also relationships between the segments formed by secants in a circle. Look closely at the following diagram. E F G H In the diagram, EG and EI are secant segments. They contain chords of the circle. The parts of the secant segments that are outside the circle are called external secant segments. In the diagram, EF and EH are external secant segments. The relationship between the secant segments can be expressed as a theorem. Theorem: If two secants intersect at a point outside a circle, the product of one secant segment and its external secant segment equals the product of the other secant segment and its external secant segment. From the diagram this means: EG EF = EI EH This theorem can be proven by drawing two auxiliary segments and using similar triangles. I Unit 4 Circles and Constructions 311

38 CTIVITY 4.4 Segment Lengths In Circles High-Quality Product SUGGESTED LERNING STRTEGIES: Think/Pair/Share 2. Give the missing reasons in the following proof. I n O T a m b P N Given: PI and PN drawn to the circle from P Prove: m n = a b Statements PI and PN drawn to the circle from P Reasons 2. Draw chords IT and ON. 2. Through any two points there is exactly one line. 3. P P OIT ONT PTI PON b n = m a mn = ab SpringBoard Mathematics with Meaning TM Geometry

39 Segment Lengths In Circles High-Quality Product CTIVITY 4.4 SUGGESTED LERNING STRTEGIES: Create Representations, Identify a Subtask EXMPLE 2 Complete the following to find DE. 8 C 5 B x D 3x E B = D = By the theorem: Whole secant segment external secant segment = whole secant segment external secant segment So, C = E nd 8 = 3x Now, = x 2 = x 2 = x = DE Unit 4 Circles and Constructions 313

40 CTIVITY 4.4 Segment Lengths In Circles High-Quality Product SUGGESTED LERNING STRTEGIES: Interactive Word Wall, Marking the Text TRY THESE B Find x in each of the following. a. 8 x 5 12 b x Finally, there are relationships between secant segments and tangent segments in a circle. Remember, a tangent segment is the segment from a point outside the circle to the point of tangency. 314 SpringBoard Mathematics with Meaning TM Geometry

41 Segment Lengths In Circles High-Quality Product CTIVITY 4.4 SUGGESTED LERNING STRTEGIES: Close Reading, Marking the Text, Prewriting, Self/Peer Revision Examine the following diagram. B C D In the diagram B is a tangent segment. The relationship between this tangent segment and the secant segments in the diagram can also be expressed as a theorem. Theorem: If a secant and a tangent intersect at a point outside a circle, the product of the length of the secant segment and its external secant segment equals the square of the length of the tangent segment. From the diagram this means: D C = B 2 gain, similar triangles can be used to prove the validity of this relationship. 3. Write a paragraph proof for the following. Hint: Use auxiliary segments NP and NO and similar triangles PNM and NOM. Given: MN and N MP drawn a to the circle from M M Prove: cb = a 2 b O P c Unit 4 Circles and Constructions 315

42 CTIVITY 4.4 Segment Lengths In Circles High-Quality Product SUGGESTED LERNING STRTEGIES: Create Representations, Identify a Subtask EXMPLE 3 Complete the following to find x. 10 D 4 x C x B BD = By the theorem: whole secant segment external secant segment = (tangent segment ) 2 So, BD = ( ) 2 nd = 10 2 Now, x 2 = 100 x 2 = x = TRY THESE C Find x in each of the following. a. b. 8 3 x 6 x SpringBoard Mathematics with Meaning TM Geometry

43 Segment Lengths In Circles High-Quality Product CTIVITY 4.4 SUGGESTED LERNING STRTEGIES: Marking the Text, Create Representations 4. Lee now believes he can find the radius of his rotor while avoiding damage to other parts. He is able to measure a segment of 5 inches across the rotor. The distance along the perpendicular bisector of the segment from its midpoint to the edge of the rotor is 1.25 inches. Use this information and what you know about special segments in circles to find the radius of the rotor to the nearest tenth of an inch in. 5 in. CHECK YOUR UNDERSTNDING 1. Find x. x 4 cm D 3. n arch over a door is 50 cm high and 200 cm wide. Find the length of the radius of the circle containing the arch. Round to the nearest tenth. 9 cm 4 cm C 50 cm 2. Find x. E 20 in. 2x 2x 200 cm 6 in. () Unit 4 Circles and Constructions 317

44 CTIVITY 4.4 Segment Lengths In Circles High-Quality Product CHECK YOUR UNDERSTNDING () 4. Find x MTHEMTICL REFLECTION Ken is trying to help Karen find x in the following diagram. Karen has used the method shown to find the value of x. x 12 x Find x. 6. Find x. 5 cm x cm 6 cm 8 cm 8(10) = 8x 80 = 8x 10 = x Is Karen s answer correct? Explain why or why not. If her answer is incorrect, what should Karen do to correct it? x 8 m 10 m 318 SpringBoard Mathematics with Meaning TM Geometry

45 ngles and Segments in Circles VERTIGO ROUND Embedded ssessment 1 Use after ctivity 4.4. The renowned architect and graduate of the MIU School of Design, Drew tower, designed a hotel, all of whose floors spin on a circular track. s it spins, each floor pauses every 45. (Otherwise, getting on and off the elevator would be tricky.) The figure below shows an overhead view of one of the square floors and its circular track. Points, B, C, and D are located at each of the four corners of the building. O bisects ZW. Z, O, and X are collinear., O, and C are collinear. 1. The circular track is tangent to each side of Quad BCD and all of the angles in Quad BCD are right angles. W, X, Y, and Z are the points of tangency. Find each of the following. a. m ZW D Z W O Y X B C b. m WXZ 2. Draw X and label the point of intersection with the circle as point M. If m ZM = 53 then find m XB. 3. Draw C and radius OX. Find each of the following. a. m OX b. m OX c. CX intercepts two arcs. Find their measures. When the building rotates 45, the corner that was located at point, is now located at point E. P, Q, R, and S are the points of tangency. 4. Draw PS and RZ. Find the measure of the angles formed by PS and RZ. P E Q B H Z F S R D C G Unit 4 Circles and Constructions 319

46 Embedded ssessment 1 Use after ctivity 4.4. ngles and Segments in Circles VERTIGO ROUND Mr. tower designed the front door of his hotel with an arch over the top. The arch is 50 cm high and the door is 180 cm wide. 50 cm 180 cm 5. To ensure the entryway is built to code, Drew needs to know the radius of the circle containing the arch. Find this radius. Explain your solution. circular stained glass and wrought iron window is to be installed above the door. The window will have to be supported underneath and above with tangent lumber or metal supports much like the diagram below. B C D 6. ssume the window and the supports will be centered over the door. If the radius of the window is 90 cm and the supports are each 200 cm long centimeters long, what is the distance from to D? Justify your answer. 320 SpringBoard Mathematics with Meaning TM Geometry

47 ngles and Segments in Circles VERTIGO ROUND Embedded ssessment 1 Use after ctivity One of the stained glass window designs being considered has several wrought iron chords each 60 cm long. How far are the chords from the center of the circle? (Remember, the radius of the window is 90 cm.) Draw a diagram and show your work. In the main lobby of the hotel, there is a circular hospitality area that also rotates. The radius of the hospitality area is 3 m. The distance from the front door, at point, to the far side of the hospitality area, at point C, is 20 m. t a certain moment the dessert bar is located at point B and the tourist information is located at point E. B D E C 8. ssuming B and E are tangent to the circle, which of the remaining distances can Drew calculate without his tape measure? Find these distances and justify your answers. Unit 4 Circles and Constructions 321

48 Embedded ssessment 1 Use after ctivity 4.4. ngles and Segments in Circles VERTIGO ROUND Exemplary Proficient Emerging Math Knowledge #2, 3a-c Finds the correct measure of angles XB, OX, and OX. (2, 3a, 3b) Finds the correct measure of the two arcs intercepted by angle CX. (3c) Finds the correct measure of only two of the angles. Finds the correct measure of only one of the arcs. Finds the correct measure of only one of the angles. Finds the correct measure of neither of the two arcs. Problem Solving #1a, b; 3c, 4, 5, 6, 7, 8 Representations #2, 3, 4, 7 Finds the correct degree measure of arcs ZW and WXZ. (1a, b) Correctly names both of the arcs intercepted by angle CX. (3c) Finds the correct measures of all of the angles formed by segments PS and RZ. (4) Correctly finds the radius of the circle containing the arch. (5) Finds the correct distance from to D (6) and finds the distance from the center of the circle to the chords. (7) Finds the remaining distances correctly. (8) Draws and labels line segments MX, C, OX, PS, and RZ correctly. (2, 3, 4) Draws and labels a diagram that correctly represents the window. (7) Finds the correct degree measure of only one of the arcs OR Uses the correct method to find the measures but makes a computational error. Correctly names only one of the arcs. Finds the correct measures of only two of the angles. Uses the correct method to find the radius but makes a computational error. Uses correct methods in the work shown to find the distances but makes a computational error. Finds only three of the remaining distances correctly. Draws and labels three or four segments correctly. Draws a correct diagram, but not all of the labels are correct. Finds the correct degree measure of neither of the angles. Correctly names neither of the arcs. Finds the correct measure of only one of the angles. Is unable to find the correct radius. Is unable to find the distances. Finds only one of the remaining distances correctly. Draws and labels one or two segments correctly. Draws an incorrect diagram. Communication #5, 6, 8 Gives a mathematically correct explanation of the method used to find the radius. (5) Gives a mathematically correct justification for the answers. (6, 8) Gives an incomplete explanation with no mathematical errors. Gives an incomplete justification with no mathematical errors. Gives an explanation that contains mathematical errors. Gives a justification that contains mathematical errors. 322 SpringBoard Mathematics with Meaning TM Geometry

49 Measures of rcs and Sectors π in the Sky SUGGESTED LERNING STRTEGIES: ctivating Prior Knowledge CTIVITY 4.5 Lance owns The Flyright Company. His company specializes in making parachutes and skydiving equipment. fter returning from a tour of Timberlake Gardens, he was inspired to create a garden in the circular drive in front of his office building. Lance decided to hire a landscape architect to design his garden. The architect told Lance that he would need to determine the area and circumference of the garden. 1. What formula can be used to determine the circumference of the circle? What information does the circumference provide about the garden? CONNECT TO HISTORY 2. What formula can be used to determine the area of the circle? What information does the area provide about the garden? The first known calculation of π ( pi ) was done by rchimedes of Syracuse ( BCE), one of the greatest mathematicians of the ancient world. rchimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons: the polygon inscribed within the circle and the polygon within which the circle was circumscribed. Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave upper and lower bounds for the area of the circle. rchimedes knew that he had not found the value of pi but only an approximation within those limits. In this way, rchimedes showed that pi is between and Unit 4 Circles and Constructions 323

50 CTIVITY 4.5 Measures of rcs and Sectors π in the Sky SUGGESTED LERNING STRTEGIES: Visualization, Create Representations The layout of The Flyright Company s building and parking lot are shown below. Office Building Parking Lot 27 ft grassy area 32 ft The dimensions of the grassy area of the parking lot are 32 ft by 27 ft. 3. What is the radius, circumference and area of the largest circle that will fit in the grassy area? Justify your answer. 324 SpringBoard Mathematics with Meaning TM Geometry

51 Measures of rcs and Sectors π in the Sky CTIVITY 4.5 SUGGESTED LERNING STRTEGIES: Think/Pair/Share, Create Representations, Identify a Subtask, Self/Peer Revision Even though larger circular gardens will fit in the grassy area, Lance decides that he would like the garden to have a diameter of 20 feet. 4. Lance decides to surround the circular garden with decorative edging. The edging is sold in 12-foot sections that can bend into curves. How many sections of the edging will Lance need to purchase to surround his garden? Justify your answer. 5. To begin building the garden, Lance needs to purchase soil. To maintain a depth of one foot throughout the garden, each bag can cover 3.5 square feet of the circle. How many bags of soil will he need to purchase? Show the calculations that lead to your answer. 6. Lance is also considering including a sidewalk around the outside of the garden, as shown below. Determine the area of the sidewalk. 20 ft 2 ft Unit 4 Circles and Constructions 325

52 CTIVITY 4.5 Measures of rcs and Sectors π in the Sky SUGGESTED LERNING STRTEGIES: Interactive Word Wall, Vocabulary Organizer, Create Representations, Quickwrite Lance would like his garden to resemble a colorful parachute with different flowers in alternating areas of the garden. His sketch for the landscaper is shown below. 7. The circle garden is divided into 8 equal parts. a. What portion of the total area of the circle is each part? b. Recall that Lance would like the diameter of his circle garden to be 20 feet. Determine the area of each sector of the garden. Show the calculations that lead to your answer. MTH TERMS sector of a circle is a region formed by two radii and an arc of a circle. CONNECT TO P You will work with sectors of circles when you study polar equations in calculus. c. What is the measure of the central angle for the sector from part a? d. Write the fraction that is equivalent to your answer in part a and that has a denominator of 360. Explain the meaning of both the numerator and denominator using circle terminology. 8. Write an equation that will allow Lance to calculate the area of a sector with a central angle of nº and radius r. 326 SpringBoard Mathematics with Meaning TM Geometry

53 Measures of rcs and Sectors π in the Sky CTIVITY 4.5 SUGGESTED LERNING STRTEGIES: Create Representations, Group Presentation 9. Determine the arc length of each sector of Lance s garden. 10. Write an equation that will allow Lance to calculate the arc length of a sector with a central angle of nº and radius r. 11. Compare and contrast the arc length of a sector and the area of a sector. 12. The architect submits his estimate to Lance. Lance notices that he has used the following formulas to calculate the area of each sector and length of each arc in the garden. degree measure of the central angle = area of sector 360 area of the circle degree measure of the central angle = arc length of a sector 360 circumference of the circle re the equations in Item 8 and Item 10 equivalent to the architect s equations? Why or why not? Unit 4 Circles and Constructions 327

54 CTIVITY 4.5 Measures of rcs and Sectors π in the Sky CHECK YOUR UNDERSTNDING Write 1. Determine your answers the circumference on notebook and paper. area of Show a your work. 7. Use circle B below to complete items a circle with a diameter of 8 inches. through d. 2. circle has circumference 12π inches. What is the area of the circle? circular rotating sprinkler sprays water over B 9 cm a distance of 9 feet. What is the area of the C circular region covered by the sprinkler? 4. Mr. Nixon boards a Ferris wheel with a. area of circle B = a diameter of 100 feet. Calculate the approximate distance that Mr. Nixon traveled b. area of shaded sector of circle B = in his 6 revolutions. Round to 3 decimal c. circumference of circle B = places. d. length of C = 5. wheel has a diameter of 24 in. How many revolutions would it take to travel 1 mile? (1 mile = 5,280 feet) 6. The length of an arc in a circle with radius 8 inches is 3.2π inches. Determine the measure of the arc. 8. The measure of the central angle of a sector is 60 and the area of the sector is 6π inche s 2. Calculate the radius of the circle. 9. MTHEMTICL REFLECTION If the area of a sector is one tenth of the area of the circle, what is the central angle of the sector? Explain how you found your answer. 328 SpringBoard Mathematics with Meaning TM Geometry

55 Equation of a Circle and a Sphere Round and Round SUGGESTED LERNING STRTEGIES: Quickwrite CTIVITY 4.6 How can the path of a windmill be described mathematically? Suppose the coordinate plane is positioned so that the center of the windmill face is at the origin. 1. Show that the points given in the diagram are on the circle by showing that the coordinates of the points satisfy the equation given for the circle. a. x 2 + y 2 = 4 y (1, 3) x CONNECT TO SCIENCE b. x 2 + y 2 = 36 (0, 2) wind farm is an array of windmills used for generating electrical power. The world s largest wind farm at ltamont Pass, California, consists of 6,000 windmills. Wind farms supply about 1.5% of California s electricity needs. y ( 5, 11) (6, 0) x Unit 4 Circles and Constructions 329

56 CTIVITY 4.6 Equation of a Circle and a Sphere Round and Round SUGGESTED LERNING STRTEGIES: Vocabulary Organizer, Create Representations, Notetaking Consider the circle at the right, which has its center at (0, 0) and has a radius of 5 units. y (x, y) MTH TERMS circle is the set of all coplanar points (x, y) that are a given distance, the radius, from a point, the center. 2. Suppose (x, y) is a point on the circle. a. Use the distance formula to write an equation to show the distance from (x, y) to (0, 0) is 5. 5 x b. Square both sides of your equation to eliminate the square root. Consider a circle that has its center at the point (h, k) and has radius r. 3. Suppose (x, y) is a point on the circle. a. Use the distance formula to write an equation to show the distance from (x, y) to (h, k) is r. y (x, y) r (h, k) x b. Square both sides of your equation to eliminate the square root. 330 SpringBoard Mathematics with Meaning TM Geometry

57 Equation of a Circle and a Sphere Round and Round CTIVITY 4.6 SUGGESTED LERNING STRTEGIES: Think/Pair/Share, Create Representations TRY THESE Write the equation of the circle described. a. center (0, 0) and radius 7 b. center (0, 2), radius =10 c. center (9, -4), radius = 15 Identify the center and radius of the circle. d. (x - 2 ) 2 = (y + 1 ) 2 = 9 e. (x + 3 ) 2 + y 2 = Consider the circle with center (2, -3) that contains the point (5, 1). a. Write the equation of any circle with radius r and center (2, -3). b. Substitute (5, 1) for (x, y) in the equation from Item 4a. Explain what information this gives about the circle. c. Write the equation of the circle with center (2, -3) that contains the point (5, 1). Unit 4 Circles and Constructions 331

58 CTIVITY 4.6 Equation of a Circle and a Sphere Round and Round SUGGESTED LERNING STRTEGIES: Think/Pair/Share, Create Representations, Quickwrite 5. Consider the circle that has a diameter with endpoints (-1, 4) and (9, -2). a. Determine the midpoint of the diameter. What information does this give about the circle? b. Using one of the endpoints of the diameter and the center of the circle, write the equation of the circle. c. Using the other endpoint of the diameter and the center of the circle, write the equation of the circle. d. What do you notice about your responses to parts b and c above? What does that tell you about writing the equation of a circle given the endpoints of the diameter? 6. Follow the steps below to write the equation of the circle that contains the three points: (-4, -1), B (6, -1), and C (6, 5). y a. Graph the three points. b. Draw the chords B and BC. c. Graph the perpendicular bisectors of the chords. d. Recall that the perpendicular bisectors of chords of a circle intersect at the center of the circle. What are the coordinates of the center of this circle? x e. Write the equation of the circle using the center and any one of the given points, B, or C. 332 SpringBoard Mathematics with Meaning TM Geometry

59 Equation of a Circle and a Sphere Round and Round CTIVITY 4.6 SUGGESTED LERNING STRTEGIES: Vocabulary Organizer, Quickwrite, Think/Pair/Share, Create Representations TRY THESE B Write the equation of the circle described. a. center (3, 6) passes through the point (0, -2) b. diameter with endpoints (2, 5) and (-10, 7) c. contains the points (-2, 5), (-2, -3), and (2, -3) sphere is the set of all points (x, y, z) that are a given distance, the radius, from a point, the center. CDEMIC VOCBULRY sphere 7. Compare and contrast the definitions of circle and sphere. The formula for the equation of a sphere is an extension of the formula for the equation of a circle. The equation of a sphere whose center is (i, j, k) and whose radius is r units long is (x - i) 2 + (y - j) 2 + (z - k) 2 = r Write the equation of a sphere whose center is at (3, -2, 4) and that has a radius of 6 units. 9. Write the equation of a sphere whose center is (0, 5, 2) and contains the point (1, 3, 8). 10. Write the equation of a sphere that has a diameter with endpoints (1, 6, -4) and (9, 6, -2) CONNECT TO P In calculus, you will compute how fast the radius and other measurements of a sphere are changing at a particular instant in time. Unit 4 Circles and Constructions 333

60 CTIVITY 4.6 Equation of a Circle and a Sphere Round and Round CHECK YOUR UNDERSTNDING Write your answers on on notebook paper. paper. Show Show your work. 4. Write an equation for the sphere described. your work. a. center (3, -2, 0) and radius = If a circle has equation (x - 3 ) + y = 4, b. center (-2, 5, -1) and contains the point which point is on the circle? (4, 5, 7) a. (1, 0) c. diameter with endpoints (2, 3, -5) and b. (2, 1) (4, 1, 5) c. (1, -2) d. (3, 0) 2. Two points on the circle given by the equation (x - 1 ) (y +3 ) = 100 have the x-coordinate -7. What are the y-coordinates of those two points? 3. Write an equation for the circle described. a. center (-3, 0) and radius = 7 b. center (4, 3), tangent to the y-axis c. center (2, -1) and contains the point (4, 5) d. diameter with endpoints ( 2, -5) and (4, 1) e. contains the points (-2, 3), (-2, 7) and (6, 3) 5. MTHEMTICL REFLECTION Suppose that you know the equation for a sphere and the coordinates of point P. How can you tell if point P lies on the sphere, inside the sphere, or outside the sphere? 334 SpringBoard Mathematics with Meaning TM Geometry

61 Constructions Constructive Thinking SUGGESTED LERNING STRTEGIES: ctivating Prior Knowledge, Close Reading, Think/Pair/Share, Use Manipulatives, Self/Peer Revision Using given definitions, theorems, and the properties of circles, geometric constructions are equivalent to visual logic problems. Historically, mathematicians posed problems to be solved using only a straightedge (a ruler without numbers) and a compass. They explored methods of constructing regular polygons. Squares, regular hexagons, and regular octagons are relatively simple to construct. Regular pentagons are more challenging. Yet, Euclid (300 BCE) demonstrated how to construct a pentagon in a circle. In this unit, eight basic constructions are introduced. From these, numerous problems can be explored from a construction perspective. Construction 1: Construct a segment congruent to a given segment. Given: EZ E Z Construct a segment congruent to EZ. CTIVITY 4.7 C C Q Use a straightedge to draw a construction segment that is clearly longer than EZ. Mark any point on the segment and label it C. Use a compass to measure EZ. Using C as the center, draw an arc on the segment. Label the intersection Q. CQ EZ This table of geometry, from the 1728 publication titled Cyclopaedia, illustrates classic constructions. Construction 2: Construct an angle congruent to a given angle. Given: CT Construct an angle congruent to CT. O Use a straightedge to draw a ray. Label the endpoint O. O G Draw an arc through the interior of CT. Using the same radius and point O as center, draw a large arc. Label the point of intersection with the ray, G. C T O D Use a compass to measure the opening of CT along the arc. Using that measure as radius and point G as center, draw an arc through the first arc. Label the intersection point D. G CDEMIC VOCBULRY geometric construction D O G Draw ray OD. DOG CT Unit 4 Circles and Constructions 335

62 CTIVITY 4.7 Constructions Constructive Thinking SUGGESTED LERNING STRTEGIES: Close Reading, Questioning the Text, Think/Pair/Share, Use Manipulatives, Quickwrite, Self/Peer Revision 1. Explain why Construction 2 guarantees that the constructed angle is congruent to the original angle. In geometry, we use the terms construct, sketch, and draw with very different, specific intent. When instructed to construct a geometric figure, it is implied that only a compass and a straightedge are permitted to complete the task. When instructed to sketch a figure, a quick jotted picture is intended to illustrate relative geometric relationships. ccurate measurements are not required in a sketch. drawing usually includes measurement with a ruler and/or a protractor. TRY THESE B C D F G Use the given figures to create the following constructions on a separate sheet of unlined paper. a. Construct and name a segment congruent to B. b. Construct and name a segment with length equal to 3CD - B. c. Construct and name an angle congruent to EFG. E d. Construct an angle with measure equal to m EFG + m HIJ e. Construct an angle with measure equal to m EFG - m HIJ I H J f. Construct PRY with PR = B, m PRY = m EFG, and RY = CD. Construction 3: Construct the perpendicular bisector of a given segment. Given: WO Construct the perpendicular bisector of WO. W G O W O W O W O W O L Using point W as center and a radius greater than half of WO (by sight), draw a large arc above and below the segment. Using the same radius and point O as center, draw a large arc to intersect the other arc in two points. Draw a line through the two intersection points. Label two points on the line. GL is the perpendicular bisector of WO. 336 SpringBoard Mathematics with Meaning TM Geometry

63 Constructions Constructive Thinking CTIVITY 4.7 SUGGESTED LERNING STRTEGIES: Close Reading, Questioning the Text, Think/Pair/Share, Use Manipulatives, Quickwrite, Self/Peer Revision 2. Explain why Construction 3 guarantees that the constructed line is the perpendicular bisector of the given segment. Construction 4: Construct the bisector of an angle. Given: Construct the bisector of. P P P R P R T T T T Using point as center, draw an arc through the angle. Label the intersection points P and T. Using P as center and an ample radius, draw an arc in the interior of the angle. Using T as center and the same radius as from P, draw an arc to intersect the previous arc. Label the intersection R. Draw R. R bisects. 3. Explain why Construction 4 guarantees that the constructed ray is the bisector of the given angle. Construction 5: Given a line and a point not on the line, construct a line through the point that is parallel to the given line. H Given: Line l and point H not on the line Construct a line through H that is parallel to line l. l There are numerous methods to construct a line parallel to a given line based on properties of angles formed by parallel lines intersected by a transversal. The construction below is called the rhombus method. H l H T l H T P l H T P l Chose any point on line l. Using point as center and radius H, draw an arc through point H and line l. Label T the point of intersection of the arc with the line. Using points H and T as center and radius H, draw arcs to locate point P, the fourth vertex of a rhombus. Draw HP. HP T Unit 4 Circles and Constructions 337

64 CTIVITY 4.7 Constructions Constructive Thinking SUGGESTED LERNING STRTEGIES: Close Reading, Questioning the Text, Think/Pair/Share, Use Manipulatives, Quickwrite, Self/Peer Revision 4. Explain why Construction 5 guarantees that the constructed line is parallel to the given line. Construction 6: Given a line and a point not on the line, construct a line through the point that is perpendicular to the given line. Given: Line l and point F not on the line Construct a line through F that is perpendicular to line l. F l F F F I N l I N l I E N l Using point F as center and a radius greater than the distance to the line, draw an arc intersecting the line in two points. Label the intersection points I and N. Use Construction 3 to construct the perpendicular bisector of IN. Label a point E on the perpendicular bisector. FE line l 5. Explain why this construction guarantees that the constructed line is perpendicular to the given line. Construction 7: Given a line and a point on the line, construct a line through the point that is perpendicular to the given line. Given: Point O on line l O Construct a line through O that is perpendicular to line l. Y l F O X l F O X l F O X l Using point O as center and a chosen radius, draw arcs on both sides of point O to intersect the line. Label the intersection points F and X. Use Construction 3 to construct the perpendicular bisector of FX. Label a point Y on the perpendicular bisector. YO line l 338 SpringBoard Mathematics with Meaning TM Geometry

65 Constructions Constructive Thinking CTIVITY 4.7 SUGGESTED LERNING STRTEGIES: Close Reading, Questioning the Text, Think/Pair/Share, Use Manipulatives, Quickwrite, Self/Peer Revision 6. Explain why Construction 7 guarantees that the constructed line is perpendicular to the given line. Construction 8: Given a line segment, divide it into two segments in a 1:2 ratio. T T Draw a segment along a diagonal from point. P I N Using a small radius, start at to mark three congruent segments along the diagonal. Label the three intersection points P, I, and N. T R P I N Draw NT. Use Construction 2 to construct PR congruent to NT. R:RT 1:2 T 7. Explain why Construction 8 guarantees that the ratio R:RT is 1:2. Unit 4 Circles and Constructions 339

66 CTIVITY 4.7 Constructions Constructive Thinking CHECK YOUR UNDERSTNDING Use the given figures to the right and unlined paper to complete your answers. Do not erase your construction marks. 1. Construct and name an isosceles triangle with legs of length EF and base of length B. Describe the steps used to complete the construction. 2. Construct and name a square with sides of length CD. Describe the steps used to complete the construction. 3. Construct and name a triangle with sides of lengths B, CD, and EF. 4. Construct and name an equilateral triangle with sides of length EF. 5. Construct and name a regular hexagon with sides of length B. Start by constructing a circle of radius B. Construct consecutive chords of length B around the circle. Explain why this construction guarantees a regular hexagon. 6. Construct and name a rectangle with consecutive sides of lengths CD and EF. 7. Construct and name a parallelogram congruent to the parallelogram given below. Describe the steps used to complete the construction. 8. Construct and name a rhombus with diagonals of lengths B and EF. 9. Construct and name a rectangle with a side of length CD and diagonals of length EF. E Z B 10. MTHEMTICL REFLECTION geometric constructions. C Y F D Compare and contrast geometric drawings with 340 SpringBoard Mathematics with Meaning TM Geometry

67 rea and rc Length, Equation of a Circle GIVE ME C-I-R-C-L-E Embedded ssessment 2 Use after ctivity 4.7. The cheerleaders at SBHS have decided to make customized megaphones to use at each of the home football games. They need to make a template to use to make the megaphones the same size. The template for the megaphone includes two concentric circles. The inner circle has a radius of 1 inch and the outer circle has a radius of 12 inches, as shown below. 1. Determine the circumference of each circle. 2. Determine the area of each circle. The cheerleaders cut along the radius drawn and cut out the inner circle for the opening at the small end of the megaphone. 3. What is the remaining area of the megaphone template? 1 in. 11 in. fter attempting to make the first megaphone from the circle template, the cheerleaders realize that they need to cut a sector (the unshaded region) out of the circle as shown at the right. 4. What is the area of the sector used to make the megaphone? 5. Once the megaphone has been assembled, assuming no overlap, what is the distance around the larger opening of the megaphone? The cheerleading coach is also the math teacher at the school, and suggests that the girls draw their circle template on a coordinate plane. 6. Positioning the center of the circles at (0, 0), write the equation of both the inner and the outer circles. 1 in. 11 in. 7. Translating the template for the megaphone so that the center of the circles is (-5, 6), write the equation of both the inner and the outer circles. Unit 4 Circles and Constructions 341

68 Embedded ssessment 2 Use after ctivity 4.7. rea and rc Length, Equation of a Circle GIVE ME C-I-R-C-L-E Exemplary Proficient Emerging Math Knowledge #1, 2, 3, 4, 6, 7 The student: Determines the correct circumferences and areas of both circles. (1, 2) Finds the correct area of the region between the circles. (3) Finds the correct area of the shaded region used to make the megaphone. (4) Determines the correct equations for both circles. (6, 7) The student: Determines the correct circumference and area of only one circle. OR Determines the correct circumferences or correct areas for both circles. Uses the correct method to find the area but makes a computational error. Uses the correct method to find the area but makes a computational error. Determines the correct equation for only one of the circles. The student: Determines only one of the four measurements. Does not find the correct area of the region. Does not find the correct area of the region. Determines the correct equation for neither of the circles. Problem Solving #5 Representations #6, 7 The student finds the correct distance around the larger opening. (5) The student: Draws the circles correctly on a coordinate plane and writes a correct equation for each of the circles. (6) Writes a correct equation for each of the translated circles. (7) The student uses a correct method to find the distance but makes a computational error. The student: Draws only one circle correctly and writes its correct equation. OR Draws both circles correctly. Writes a correct equation for only one of the translated circles. The student does not find the correct distance. The student: Draws only one circle correctly. Writes a correct equation for neither of the translated circles. 342 SpringBoard Mathematics with Meaning TM Geometry

69 Practice UNIT 4 CTIVITY In the diagram below, BC = 16 in. and PC = 12 in. Calculate B. B 6. Solve for x in the following figures. a. Given: C is the center of the circle. 12x + 6 3x 6 P C C 2. chord of a circle is 12 in. long, and its midpoint is 8 in. from the center of the circle. Calculate the length of the radius of the circle. 3. In a circle with radius 15 cm, SB is a chord that has length 24 cm. How far is SB from the center of the circle? 4. In the diagram, B = 8 in., C = 14 in., CE = 10 in., and GE = 8 in. Determine the length of G. B C b x 120 3x 1 7. PQ is tangent to the circle. m 1 = t + 3. Solve for t. H G F D E 10t 26 1 P Q a. 4 b. 5 c. 8 d. 11 CTIVITY Given a circle with center C and m SCR = 42. Determine each of the following. a. m SR b. m SCQ S c. m SQR d. m Q C R QV e. m QSV V f. m QVR 8. Draw a sketch of a regular pentagon inscribed in a circle. Find the measure of each arc intercepted by the sides of the pentagon. 9. Given: B is a diameter. Explain why 1, 2, and 3 are all right angles B Unit 4 Circles and Constructions 343

70 UNIT 4 Practice CTIVITY Use the diagram below to answer each of the following. b CTIVITY Find x. x 18 8 x a 1 c 2 d 14. Find x. a. If a = 120, c = 84 and d = 108, then m 1 = and m 2 =. b. If a = 110, and m 1 = 88, then c = and b + d =. 11. Use the diagram below to answer each of the following. 4 x Find x. Q 9 N M R a. If m MPQ = 48, then m QM =. b. If m QNM = 200, then m MPQ =. c. If m QN = 125 and m QR = 83, then m 1 =. d. If m QNR = 260 and m 1 = 45, then m QN =. 12. Use always, sometimes, or never to make each of the following statements true. a. parallelogram inscribed in a circle is a rectangle. b. n inscribed angle that intercepts an arc whose measure is greater than 180 is acute. c. If two angles intercept the same arc, they are congruent. 1 P 16. Find x. 5 CTIVITY 4.5 x 2 x 17. circle has area 169π in. 2 Determine the circumference of the circle. 18. Determine the area of a circle that has a circumference of 16π cm. 19. The measure of the arc of a sector is 72 and the 2 area of the sector is 5π in. Calculate the radius of the circle SpringBoard Mathematics with Meaning TM Geometry

71 Practice UNIT Determine the area of the shaded region a. 3 in. b in. 2 in. B 21. Using the circle in Item 20b, determine the length of B. CTIVITY Write an equation for the circle described. a. center (0, -3), radius = 5 b. center (5, 5), tangent to both axes. c. center (1, -2), tangent to the x-axis. d. center (0, 6), passes through (6, 14) e. diameter with endpoints (-3, 4) and (3, -4) f. diameter with endpoints (-1, 5) and (3, 7) g. contains the points (1, 4), (9, 4) and (9, -2) 23. Identify the center and radius of the circle given by each equation. a. (x - 3) 2 + y 2 = 1 b. (x + 2) 2 + (y + 1) 2 = Two points on the circle (x - 2) 2 + (y - 4) 2 = 25 both have the y-coordinate 7. What are the x-coordinates of those two points? 25. Write an equation for the sphere described. a. center (-5, 0, 4) and radius = 11 b. center (10, -6, 2) and contains the point (10, -1, -10) c. diameter with endpoints (-8, 1, -7) and (6, 3, -1) CTIVITY 4.7 Use the given figures below and unlined paper to complete your answers. Do not erase your construction marks. 26. Construct and name a triangle with angles congruent to and B and the included side of length HI. 27. Construct and name an isosceles trapezoid with base of length JK, base angles congruent to B and legs of length FG. 28. Construct and name a right triangle with legs of lengths HI and JK. 29. Construct and name a regular hexagon with sides of length FG. F H J B G I K Unit 4 Circles and Constructions 345