4.4 Perpendicular Lines

Transcription

1 OMMON ORE Locker LESSON 4.4 erpendicuar Lines Name ass ate 4.4 erpendicuar Lines ommon ore Math Standards The student is expected to: OMMON ORE G-O..9 rove theorems about ines and anges. so G-O..12 Mathematica ractices OMMON ORE M.5 Using Toos Language Objective Expain to a partner why a pair of ines is or is not perpendicuar. ENGGE Essentia Question: What are the key ideas about perpendicuar bisectors of a segment? ossibe answer: If you know that a ine is the perpendicuar bisector of a segment, then any point on the ine is equidistant from the endpoints of the segment. If you know that a point on a ine is equidistant from the endpoints of a segment, then the ine must be the perpendicuar bisector of the segment. REVIEW: LESSON ERFORMNE TSK View the Engage section onine. iscuss the photo. sk students how they woud baance the competing factors that a power company might face in deciding where to buid a wind farm. Then preview the Lesson Houghton Miffin Harcourt ubishing ompany Essentia Question: What are the key ideas about perpendicuar bisectors of a segment? 1 Expore onstructing erpendicuar isectors and erpendicuar Lines You can construct geometric figures without using measurement toos ike a ruer or a protractor. y using geometric reationships and a compass and a straightedge, you can construct geometric figures with greater precision than figures drawn with standard measurement toos. In Steps, construct the perpendicuar bisector of. ace the point of the compass at point. Using a compass setting that is greater than haf the ength of, draw an arc. Without adjusting the compass, pace the point of the compass at point and draw an arc intersecting the first arc in two paces. Labe the points of intersection and. In Steps E, construct a ine perpendicuar to a ine that passes through some point that is not on. ace the point of the compass at. raw an arc that intersects ine at two points, and. E Use the methods in Steps to construct the perpendicuar bisector of. ecause it is the perpendicuar bisector of, then the constructed ine through is perpendicuar to ine. Resource Locker Use a straightedge to draw, which is the perpendicuar bisector of. erformance Task. 195 Modue Lesson 4 Name ass ate 4.4 erpendicuar Lines Essentia Question: What are the key ideas about perpendicuar bisectors of a segment? 1 Expore onstructing erpendicuar isectors Houghton Miffin Harcourt ubishing ompany and erpendicuar Lines You can construct geometric figures without using measurement toos ike a ruer or a protractor. y using geometric reationships and a compass and a straightedge, you can construct geometric figures with greater precision than figures drawn with standard measurement toos. In Steps, construct the perpendicuar bisector of Ā. G-O..9 rove theorems about ines and anges. so G-O..12 ace the point of the compass at point. Using a compass setting that is greater than haf the ength of, draw an arc. Without adjusting the compass, pace the point of the compass at point and draw an arc intersecting the first arc in two paces. Labe the points of intersection and. In Steps E, construct a ine perpendicuar to a ine that passes through some point that is not on. ace the point of the compass at. raw an arc that intersects ine at two points, and. Use the methods in Steps to construct the perpendicuar bisector of. Resource Use a straightedge to draw, which is the perpendicuar bisector of. HROVER GES Turn to these pages to find this esson in the hardcover student edition. ecause it is the perpendicuar bisector of, then the constructed ine through is perpendicuar to ine. Modue Lesson Lesson 4.4

2 Refect 1. In Step of the first construction, why do you open the compass to a setting that is greater than haf the ength of? This ensures that the two arcs wi intersect at two points. 2. What If? Suppose Q is a point on ine. Is the construction of a ine perpendicuar to through Q any different than constructing a perpendicuar ine through a point not on the ine, as in Steps and E? onstructing the points and on ine is different in the two constructions. For a point Q on ine, you pace the compass point at Q and draw arcs on either side of Q. The intersection points wi be points and. Then, you can foow the same methods as in Steps in the Expore. Expain 1 roving the erpendicuar isector Theorem Using Refections You can use refections and their properties to prove a theorem about perpendicuar bisectors. These theorems wi be usefu in proofs ater on. erpendicuar isector Theorem If a point is on the perpendicuar bisector of a segment, then it is equidistant from the endpoints of the segment. Exampe 1 Refect rove the erpendicuar isector Theorem. Given: is on the perpendicuar bisector m of. rove: = m onsider the refection across ine m. Then the refection of point across ine m is aso because point ies on ine m, which is the ine of refection. so, the refection of point across ine m is by the definition of refection. Therefore, = because refection preserves distance. 3. iscussion What concusion can you make about KLJ in the diagram using the erpendicuar isector Theorem? K M L JK = JL because point J ies on the perpendicuar bisector of KL. J Modue Lesson 4 Houghton Miffin Harcourt ubishing ompany EXLORE onstructing erpendicuar isectors and erpendicuar Lines INTEGRTE TEHNOLOGY Students have the option of exporing the perpendicuar bisector activity either in the book or onine. QUESTIONING STRTEGIES Why do you have to use a compass setting greater than haf the ength of the segment when you construct the perpendicuar bisector of the segment? This ensures that the two arcs wi intersect. INTEGRTE MTHEMTIL RTIES Focus on ritica Thinking M.3 Have students think about the steps used with each construction. sk them to refect on how they remember how to do each. In sma groups, have students discuss strategies for remembering the steps. Then share each group s best strategies with the cass. EXLIN 1 roving the erpendicuar isector Theorem Using Refections ROFESSIONL EVELOMENT Integrate Mathematica ractices This esson provides an opportunity to address Mathematica ractice M.5, which cas for students to use appropriate toos. They begin the esson by constructing the perpendicuar bisector of a segment; they may aso use paperfoding or refective devices to construct the perpendicuar bisector. Students aso construct the perpendicuar to a ine through a point not on the ine. For each construction, they must be abe to use the toos in a variety of ways in order to obtain accurate resuts and to understand the underying mathematica reationships. QUESTIONING STRTEGIES Suppose you construct the perpendicuar bisector of a segment and then choose any point on the perpendicuar bisector. If you measure the distance from the point to each endpoint of the segment, what do you expect to find? The distances from the point to each endpoint of the segment are equa. erpendicuar Lines 196

3 VOI OMMON ERRORS When finding the distance from a point on one side of a perpendicuar bisector to its refected point, students may give the distance to the bisector as the soution. Remind students to re-read the question to verify that they have answered it by using given information, such as congruence markings. EXLIN 2 roving the onverse of the erpendicuar isector Theorem Your Turn Use the diagram shown. is the perpendicuar bisector of. 4. Suppose E = 16 cm and = 20 cm. Find. E ecause is the perpendicuar bisector of, then = and = 20 cm. 5. Suppose E = 15 cm and = 25 cm. Find. ecause is the perpendicuar bisector of, then = and = 25 cm. Expain 2 roving the onverse of the erpendicuar isector Theorem The converse of the erpendicuar isector Theorem is aso true. In order to prove the converse, you wi use an indirect proof and the ythagorean Theorem. In an indirect proof, you assume that the statement you are trying to prove is fase. Then you use ogic to ead to a contradiction of given information, a definition, a postuate, or a previousy proven theorem. You can then concude that the assumption was fase and the origina statement is true. a c a 2 + b 2 = c 2 Reca that the ythagorean Theorem states that for a right triange with egs of ength a and b and a hypotenuse of ength c, a 2 + b 2 = c 2. INTEGRTE MTHEMTIL RTIES Focus on ommunication M.3 Make sure students understand that the proof of the onverse of the erpendicuar isector Theorem is based on the method of indirect proof. To write an indirect proof: 1. Identify the conjecture to be proven. 2. ssume the opposite of the concusion is true. 3. Use direct reasoning to show the assumption eads to a contradiction. 4. oncude that since the assumption is fase, the origina conjecture must be true. QUESTIONING STRTEGIES How can you te that an indirect proof is used? In the first step, you assume that what you are trying to prove is fase. Near the end of an indirect proof, a step contradicts a known true statement. What does this mean in terms of the proof? The origina assumption is fase, so what you are trying to prove must be true. Houghton Miffin Harcourt ubishing ompany onverse of the erpendicuar isector Theorem If a point is equidistant from the endpoints of a segment, then it ies on the perpendicuar bisector of the segment. Exampe 2 b rove the onverse of the erpendicuar isector Theorem Given: = rove: is on the perpendicuar bisector m of. Step : ssume what you are trying to prove is fase. ssume that is not on the perpendicuar bisector m of. Q Then, when you draw a perpendicuar ine from to the ine containing and, it intersects at point Q, which is not the midpoint of. Step : ompete the foowing to show that this assumption eads to a contradiction. Q forms two right trianges, Q and Q. So, Q 2 + Q 2 = 2 and Q 2 + Q 2 = 2 by the ythagorean Theorem. Subtract these equations: Q 2 + Q 2 = Q + Q = 2 Q Q = However, 2-2 = 0 because =. Therefore, Q Q = 0. This means that Q 2 2 = Q and Q = Q. This contradicts the fact that Q is not the midpoint of. Thus, the initia assumption must be incorrect, and must ie on the perpendicuar bisector of. Modue Lesson 4 OLLORTIVE LERNING Whoe ass ctivity Have groups of students create posters to describe how to do an indirect proof. Then demonstrate the method for the onverse of the erpendicuar isector Theorem. sk them to dispay their resuts as a graphic organizer that shows the steps for the proof. m 197 Lesson 4.4

4 Refect 6. In the proof, once you know Q 2 2 = Q, why can you concude that Q = Q? Take the square root of both sides. Since distances are nonnegative, Q = Q. Your Turn 7. is 10 inches ong. is 6 inches ong. Find the ength of. Since is equidistant from and and is perpendicuar to by the diagram, then must be the perpendicuar bisector of and = = 10, so = 8 in. and = 16 in. Expain 3 roving Theorems about Right nges The symbo means that two figures are perpendicuar. For exampe, m or XY. Exampe 3 rove each theorem about right anges. If two ines intersect to form one right ange, then they are perpendicuar and they intersect to form four right anges. Given: m 1 = 90 rove: m 2 = 90, m 3 = 90, m 4 = OMMUNITING MTH Hep students understand the onverse of the erpendicuar isector Theorem by having them make a poster showing the theorem and its converse. Then have them expain what is given as the hypothesis of the theorem and of its converse, and what they have to prove for the theorem and its converse. EXLIN 3 roving Theorems bout Right nges Statement 1. m 1 = Given 2. 1 and 2 are a inear pair. 2. Given 3. 1 and 2 are suppementary. 3. Linear air Theorem Reason 4. m 1 + m 2 = efinition of suppementary anges m 2 = Substitution roperty of Equaity 6. m 2 = Subtraction roperty of Equaity 7. m 2 = m 4 7. Vertica nges Theorem 8. m 4 = Substitution roperty of Equaity 9. m 1 = m 3 9. Vertica nges Theorem 10. m 3 = Substitution roperty of Equaity If two intersecting ines form a inear pair of anges with equa measures, then the ines are perpendicuar. Given: m 1 = m m rove: m y the diagram, 1 and 2 form a inear pair so 1 and 2 are suppementary by the Linear air Theorem. y the definition of suppementary anges, m 1 + m 2 = 180. It is aso given that m 1 = m 2, so m 1 + m 1 = 180 by the Substitution roperty of Equaity. dding gives 2 m 1 = 180, and m 1 = 90 by the ivision roperty of Equaity. Therefore, 1 is a right ange and m by the definition of perpendicuar ines. Modue Lesson 4 Houghton Miffin Harcourt ubishing ompany QUESTIONING STRTEGIES If a inear pair of anges has equa measure, why are the anges right anges? Since a inear pair of anges is suppementary, their sum must be 180. So each ange must be haf of 180, or 90. INTEGRTE MTHEMTIL RTIES Focus on Technoogy M.5 Encourage students to use the geometry software to create inear pairs of anges and then measure them. Then have them construct perpendicuar ines and measure each of the anges to verify that they are right anges. IFFERENTITE INSTRUTION Visua ues Use coors to identify the steps in an indirect proof. Write each of the steps in a different coor. Then pace coored boxes around the parts of the proof that correspond to each step. For exampe, write the step ssume the opposite of the concusion is true in bue and put a bue box around the assumption made in the indirect proof. erpendicuar Lines 198

5 ELORTE QUESTIONING STRTEGIES How is constructing a perpendicuar bisector reated to constructing a segment bisector? The construction of a perpendicuar bisector starts with a segment. How is constructing a perpendicuar bisector reated to constructing a perpendicuar to a point on a ine? The construction of a perpendicuar to a point on a ine starts with marking equa distances from the point aong the ine. This creates a segment with the point as the midpoint of the segment. Refect 8. State the converse of the theorem in art. Is the converse true? If two intersecting ines are perpendicuar, then they form a inear pair of anges with equa measures; yes. Your Turn 9. Given: b ǁ d, c ǁ e, m 1 = 50, and m 5 = 90. Use the diagram to find m 4. d b a 1 c e Eaborate m 4 = 40 ; by corresponding anges because c ǁ e, m 1 = m 2, and by vertica anges, m 2 = m 3, so m 3 = 50 ; because m 5 = 90, then a d and m 3 + m 4 = 90, so m 4 = 40. SUMMRIZE THE LESSON If you are given a ine and a point, how do you construct a ine that is perpendicuar to the given ine using a compass and straightedge? Sampe answer: Use a compass to ocate two points and on the ine that are the same distance from point. Then construct the perpendicuar bisector of. Houghton Miffin Harcourt ubishing ompany 10. iscussion Expain how the converse of the erpendicuar isector Theorem justifies the compass-and-straightedge construction of the perpendicuar bisector of a segment. The construction invoves making two arcs that intersect in two points. Each of these two intersection points is equidistant from the endpoints of the segment, because the arcs are the same radius. So, both of the intersection points are on the perpendicuar bisector of the segment. 11. Essentia Question heck-in How can you construct perpendicuar ines and prove theorems about perpendicuar bisectors? onstructing a ine perpendicuar to a given ine invoves using a compass to ocate two points that are not on the given ine but are equidistant from two points on the given ine. You can prove the erpendicuar isector Theorem using a refection and its properties, and you can prove the onverse of the erpendicuar isector Theorem using an indirect argument invoving the ythagorean Theorem. Modue Lesson 4 LNGUGE SUORT Visua ues Have students work in pairs. Give students pictures of intersecting, parae, skew, and perpendicuar ines. Instruct one student in each pair to expain why a pair of ines is or is not perpendicuar, and prove it to the partner. Have the student who is not expaining write notes about the proof expanation. Then have students switch roes. 199 Lesson 4.4

6 Evauate: Homework and ractice 1. How can you construct a ine perpendicuar to ine that passes through point using paper foding? Fod ine onto itsef so that the crease passes through point. Onine Homework Hints and Hep Extra ractice EVLUTE The crease is the required perpendicuar ine. 2. heck for Reasonabeness How can you use a ruer and a protractor to check the construction in Eaborate Exercise 10? Use the ruer to check that bisects. Use the protractor to check that and are perpendicuar. 4. Represent Rea-Word robems fied of soybeans is watered by a rotating irrigation system. The watering arm,, rotates around its center point. To show the area of the crop of soybeans that wi be watered, construct a circe with diameter. Use the diagram to find the engths. is the perpendicuar bisector of. Q is the perpendicuar bisector of. = =. 3. escribe the point on the perpendicuar bisector of a segment that is cosest to the endpoints of the segment. The midpoint of the segment is the point on the perpendicuar bisector that is cosest to the endpoints of the segment. 5. Suppose = 5 cm. What is the ength of? 6. Suppose = 5 cm and Q = 8 cm. What is the ength of Q? y the erpendicuar isector Theorem, = 5 cm. y the erpendicuar isector Theorem, Q = 8 cm. Q Houghton Miffin Harcourt ubishing ompany Image redits: sima/ Shutterstock SSIGNMENT GUIE oncepts and Skis Expore onstructing erpendicuar isectors and erpendicuar Lines Exampe 1 roving the erpendicuar isector Theorem Exampe 2 roving the onverse of the erpendicuar isector Theorem Exampe 3 roving Theorems about Right nges VOI OMMON ERRORS ractice Exercises 1 4 Exercises 5 8, Exercises 9 11, 14 Exercises 12 13, common error when working with perpendicuar ines is to assume that ines are perpendicuar if they ook perpendicuar. Emphasize the need to estabish that there are right anges before saying that ines are perpendicuar. Modue Lesson 4 Exercise epth of Knowedge (.O.K.) OMMON ORE Mathematica ractices 1 1 Reca of Information M.5 Using Toos Skis/oncepts M.5 Using Toos Skis/oncepts M.6 recision 14 3 Strategic Thinking M.6 recision Skis/oncepts M.6 recision 17 3 Strategic Thinking M.2 Reasoning 18 3 Strategic Thinking M.3 Logic 19 3 Strategic Thinking M.2 Reasoning erpendicuar Lines 200

7 INTEGRTE MTHEMTIL RTIES Focus on ritica Thinking M.3 sk students to compare constructions using a refective device, tracing paper, compass and straightedge, and geometry software. iscuss the advantages of each. Have students consider not ony how easy each is to use, but aso how accurate, and how we each too heps them understand the geometric reationships being studied. INTEGRTE MTHEMTIL RTIES Focus on Technoogy M.5 sk each student to draw a simpe sketch that invoves perpendicuar ines and right anges. Have students exchange sketches and attempt to reproduce the one they receive using geometry software. 7. Suppose = 12 cm and Q = 10 cm. What is the ength of Q? Q = 8 cm; = 12 cm so = 1 2 = 6 cm and = 6 cm. y the ythagorean Theorem, Q 2 = Q 2 + 2, so = Q and Q = 8 cm. 8. Suppose = 3 cm and = 12 cm. What is the ength of? Given: = and =. Use the diagram to find the engths or ange measures described. 9. Suppose m 2 = 38. Find m 1. m 1 = 52 ; because = and =, then is the perpendicuar bisector of, and m = 90. Then m 1 = = Suppose = 10 cm and = 6 cm. What is the ength of? = 16 cm; because = and =, then is the perpendicuar bisector of and is a right triange. y the ythagorean Theorem, 2 = 2 + 2, so = and = 8 cm. Then = 2 = 16 cm. = 5 cm; = 12 cm, so = = = 4 cm. y the ythagorean Theorem, 2 = 2 + 2, so 2 = and = 5 cm. 11. Find m 3 + m m 3 + m 4 = 90 ; because and are equidistant from the endpoints, then is the perpendicuar bisector of. So, and the ines meet at right anges, and m 3 + m 4 = 90. Given: m ǁ n, x ǁ y, and y m. Use the diagram to find the ange measures. Houghton Miffin Harcourt ubishing ompany m n a x y Suppose m 7 = 30. Find m Suppose m 1 = 90. What is m 2 + m 3 + m 5 + m 6? y m, so m 7 + m 8 = 90 and m 8 = 60. ecause m 1 = 90, then x n and the Using aternate interior anges, because x ǁ y, ines intersect at four right anges. So, then m 8 = m 6, so m 6 = and 3 m 2 + m 3 + m 5 + m 6 = 180. are vertica anges, so m 3 = 60. Modue Lesson Lesson 4.4

8 Use this diagram of trusses for a rairoad bridge in Exercise 14. E F Suppose E is the perpendicuar bisector of F. Which of the foowing statements do you know are true? Seect a that appy. Expain your reasoning.. = F INTEGRTE MTHEMTIL RTIES Focus on Math onnections M.1 Remind students about how to use the erpendicuar isector Theorem and its converse to write equations that wi hep them find ange measures reated to perpendicuar ines.. m 1 + m 2 = 90. E is the midpoint of F.. m 3 + m 4 = 90 E.,, and ; the given information that E is the perpendicuar bisector of F means that both points E and are equidistant from and F so answer choices and are true. Then, E F and m 3 + m 4 = 90, so answer choice is known to be true. The given information does not te anything about or, though, so answer choices and E may or may not be true. 15. gebra Two ines intersect to form a inear pair with equa measures. One ange has the measure 2x and the other ange has the measure (20y - 10). Find the vaues of x and y. Expain your reasoning. x = 45, y = 5; the inear pair formed has equa measures, so the ines are perpendicuar and then 2x = 90, so x = 45, and (20y - 10) = 90, so y = 5. INTEGRTE MTHEMTIL RTIES Focus on Modeing M.4 Have students expain how they can use a protractor or refective device to check the accuracy of their constructions. 16. gebra Two ines intersect to form a inear pair of congruent anges. The measure of one ange is (8x + 10) and the measure of the other ange is 15y ( ). Find the vaues 2 of x and y. Expain your reasoning. x = 10, y = 12; the inear pair formed are congruent anges, so the ines are perpendicuar and then (8x + 10) = 90, so x = 10, and ( 15y 2 ) = 90, so y = 12. H.O.T. Focus on Higher Order Thinking 17. ommunicate Mathematica Ideas The vave pistons on a trumpet are a perpendicuar to the ead pipe. Expain why the vave pistons must be parae to each other. The vave pistons are ines that are perpendicuar to the same ine (the ead pipe), so they form right anges with the same ine. y the converse of the corresponding anges theorem, a the congruent right anges mean the vave pistons are parae to each other. ead pipe vave pistons Houghton Miffin Harcourt ubishing ompany Modue Lesson 4 erpendicuar Lines 202

9 OLLORTIVE LERNING TIVITY Have students work in groups of three or four. sk them to choose one student to give directions. Instruct the other students to each draw a ine and a point that is not on the ine. The first student wi then give a set of directions, step by step, for constructing a perpendicuar to a ine through a point that is either on the ine or not on the ine. The other students wi not know which construction it is unti they have foowed the directions. Once the student has successfuy guided the other students through the construction process, have another student take the ead and describe the construction to the others. 18. Justify Reasoning rove the theorem: In a pane, if a transversa is perpendicuar to one of two parae ines, then it is perpendicuar to the other. Given: RS and Statements 1. ǁ 2. m RT = m RV 3. RS 4. m RT = m RV = RS rove: RS Reasons 19. nayze Mathematica Reationships ompete the indirect proof to show that two suppementary anges cannot both be obtuse anges. Given: 1 and 2 are suppementary. 1. Given rove: 1 and 2 cannot both be obtuse. 2. orresponding nges Theorem 3. Given 4. efinition of perpendicuar ines 5. Substitution roperty of Equaity 6. efinition of perpendicuar ines ssume that two suppementary anges can both be obtuse anges. So, assume that R T V S JOURNL Have students draw and mark a set of figures to iustrate the erpendicuar isector Theorem and its converse. Have students provide expanatory captions for the figures. Houghton Miffin Harcourt ubishing ompany 1 and 2 are both obtuse. Then m 1 > 90 and m 2 > 90 by the definition of obtuse anges. dding the two inequaities, m 1 + m 2 > 180. However, by the definition of suppementary anges, m 1 + m 2 = 180 information. This means the assumption is 1 and 2 cannot both be obtuse. So m 1 + m 2 > 180 contradicts the given fase, and therefore. Modue Lesson Lesson 4.4

10 Lesson erformance Task utiity company wants to buid a wind farm to provide eectricity to the towns of cton, axter, and oevie. ecause of concerns about noise from the turbines, the residents of a three towns do not want the wind farm buit cose to where they ive. The company comes to an agreement with the residents to buid the wind farm at a ocation that is equay distant from a three towns. cton VOI OMMON ERRORS Students can use proportions to find the actua distances between towns, but they may set them up incorrecty. To find the distance between cton and oevie, they can write and sove this proportion: 1 in. 10 mi = 1.5 in.. The correct pattern is x mi map distance actua distance = map distance actua distance. 1.5 in. 120 oevie 4 in. Scae 1 in. : 10 mi a. Use the drawing to draw a diagram of the ocations of the towns using a scae of 1 in. : 10 mi. raw the 4-inch and 1.5-inch ines with a 120 ange between them. Write the actua distances between the towns on your diagram. b. Estimate where you think the wind farm wi be ocated. axter c. Use what you have earned in this esson to find the exact ocation of the wind farm. What is the approximate distance from the wind farm to each of the three towns? a. istances: cton axter, about 49 mies; axter oevie, 40 mies; cton oevie, 15 mies b. Student answers wi vary. c. Students shoud construct perpendicuar bisectors of the three ines of the triange. The point of intersection of the bisectors is the point that is equidistant from the three vertices of the triange. ny two of the bisectors is sufficient to ocate the point, but a third one is usefu for spotting possibe errors in the drawing of the first two bisectors. Houghton Miffin Harcourt ubishing ompany INTEGRTE MTHEMTIL RTIES Focus on ritica Thinking M.3 Marcus said that it isn t necessary to draw three perpendicuar bisectors to find the ocation of the wind farm. Two is sufficient, he said. Was he right? Expain. Yes; sampe answer: ny two of the perpendicuar bisectors wi intersect at a point. That point must be the ocation of the wind farm because the two ines can intersect in ony one point. The third perpendicuar bisector provides a usefu check on the accuracy of the constructions of the first two perpendicuar bisectors. Modue Lesson 4 EXTENSION TIVITY ose this chaenge to students: Three other towns have signed up to obtain eectricity from the wind farm. are the same distance from the farm as cton, axter, and oevie. On your drawing, show three points that coud be the ocations of the towns. Expain how you found the points and how you know they meet the conditions of the probem. Students shoud draw a circe with its center at the wind farm and passing through cton, axter, and oevie. They can choose any three points on the circe as the ocations of the three new towns. Those points must be the same distance from the farm as cton, axter, and oevie because a radii of a circe are equa in ength. Scoring Rubric 2 points: Student correcty soves the probem and expains his/her reasoning. 1 point: Student shows good understanding of the probem but does not fuy sove or expain his/her reasoning. 0 points: Student does not demonstrate understanding of the probem. erpendicuar Lines 204