1.3 Points, Lines, and Planes

Transcription

1 1.3 oints, ines, and lanes oal Use postulates and undefined terms. ey Words undefined term point, line, plane postulate collinear, coplanar segment ray he legs of the tripod touch the table at three points. he legs suggest lines, and the table surface suggests a plane. eometry depends on a common understanding of terms such as point, line, and plane. ecause these terms cannot be mathematically defined using other known words, they are called undefined terms. point has no dimension. It is represented by a small dot. point line has one dimension. It extends without end in two directions. It is represented by a line with two arrowheads. line l or ^&*( plane has two dimensions. It is represented by a shape that looks like a floor or wall. You have to imagine that it extends without end. l plane or plane You need two points to describe a line, and you need three points to describe a plane, because the geometry in this book follows the two postulates given below. ostulates are statements that are accepted without further justification. OU 1 and 2 ostulate 1 wo oints etermine a ine n Words hrough any two points there is exactly one line. ymbols ine n passes through points and Q. ostulate 2 hree oints etermine a lane Words hrough any three points not on a line there is exactly one plane. ymbols lane passes through points,, and. 14 hapter 1 asics of eometry

2 X 1 Name oints, ines, and lanes Use the diagram at the right. a. Name three points. b. Name two lines. c. Name two planes. olution Πm p a.,, and are points. b. ine m and line p c. Q and are planes. ollinear points oplanar points are points that lie on the same line. are points that lie on the same plane. oplanar lines are lines that lie on the same plane. Visualize It! In xample 2 the points,,, and are also coplanar. he plane containing them is shown in green above. X 2 Name ollinear and oplanar oints Use the diagram at the right. a. Name three points that are collinear. b. Name four points that are coplanar. c. Name three points that are not collinear. olution a. oints,, and lie on the same line. o, they are collinear. b. oints,,, and lie on the same plane, so they are coplanar. c. oints,, and do not lie on the same line. here are many other correct answers. Name oints, ines, and lanes Use the diagram at the right. 1. Name two lines. 2. Name two planes. n 3. Name three points that are collinear. 4. Name three points that are not collinear. 5. Name four points that are coplanar. p m 6. Name two lines that are coplanar. 1.3 oints, ines, and lanes 15

3 tudent elp VIU Y dd these words to your vocabulary pages in your notebook, as shown on p. 2. he line ^&( passes through and. he segment &* consists of the on ^&( that are between and. s and, and all points he ray &( consists of the and all points on ^&( that lie on the same side of as. UY IN, N, and Y Word ymbol iagram line ^&( or ^&( segment &* or &* ray &( &( Note that ^&( is the same as ^&(. lso, &* is the same as &*. owever, &( is not the same as &(. he two rays have different s and extend in different directions. X 3 raw ines, egments, and ays raw three noncollinear points,,, and. hen draw ^&(, &*, and **(. olution 1 raw,, and. 2 raw ^&(. 3 raw &*. 4 raw &(. raw ines, egments, and ays 7. raw four points as shown. 8. raw the lines ^&( and ^&(. re the lines the same? xplain. 9. raw the line segments &* and &*. re the segments the same? xplain. 10. raw the rays &( and &(. re the rays the same? xplain. 16 hapter 1 asics of eometry

4 1.3 xercises uided ractice Vocabulary heck 1. Write in words how you would say each of these symbols aloud: Q ^&(, Q&*, Q&(, and Q &(. 2. xplain the difference between Q&( and Q &(. kill heck ecide whether the statement is true or false. 3. oints,, and are collinear. 4. oints,, and are coplanar. 5. oints,, and are coplanar. 6. oint lies on ^&(. 7. ^&( lies on plane. 8. ^&( lies on plane. ketch a line that contains point between points and. ecide whether the statement is true or false. 9. ^&( is the same as ^&(. 11. &( is the same as &(. 13. &* is the same as &*. 10. &( is the same as &(. 12. &( is the same as &(. 14. &* is the same as &*. ractice and pplications xtra ractice Naming oints, ines, and lanes Use the diagram at the right. ee p Name four points. 16. Name two lines. 17. Name the plane that contains,, and. 18. Name the plane that contains,, and. valuating tatements ecide whether the statement is true or false. omework lp xample 1: xs xample 2: xs xample 3: xs lies on line l. 20.,, and are collinear. 21. lies on line l. 22.,, and are coplanar. 23. lies on line m. 24.,, and are collinear. 25. lies on line m. 26.,, and are coplanar. m l 1.3 oints, ines, and lanes 17

5 Naming ollinear oints Name a point that is collinear with the given points. 27. and 28. and 29. and 30. and N 31. and N 32. and 33. and 34. and Naming Noncollinear oints Name three points that are not collinear U V N W Z X Y Itudent elp I ZON.O OWO xtra help with problem solving in xs is at classzone.com Naming oplanar oints Name a point that is coplanar with the given points. 38.,, and 39.,, and 40.,, and 41.,, and 42.,, and 43.,, and 44.,, and 45.,, and Naming Noncoplanar oints Name all the points that are not coplanar with the given points. 46. N,, and 47.,, and 48., Q, and N 49.,, and 50., Q, and 51.,, and N 52.,, and 53. Q,, and N ame oard In xercises 54 57, use the game board. 54. Name four collinear points. 55. Name three points that are not collinear. 56. Name four segments that contain point. 57. ^&( divides the board in half. Q ^&( also divides the board in half. Name the other lines that divide the board in half. U Q V 18 hapter 1 asics of eometry

6 You be the udge indoor tennis court. 58. Name two points that are collinear with. 59. Name three points that are coplanar with. 60. Name two planes that contain. 61. Name two planes that do not contain. 62. re the points and N coplanar with points and Q? xplain. In xercises 58 62, use the diagram of the ketch the lines, segments, and rays. If you have geometry software, try creating your sketch using it. 63. raw four points,,, and, no three of which are collinear. ketch ^**(, &*, ^&(, and &(. 64. raw two points, and. ketch &(. dd a point on the ray so is between and. 65. raw three noncollinear points,, and. ketch &* and add a point on &*. hen sketch &(. hree-wheeled ar In xercises 66 69, refer to the photograph of the three-wheeled car. 66. four-wheeled car is driving slowly over uneven ground. Is it possible that only three wheels will be touching the ground at a given time? 67. Is it possible to draw four points that do not lie on a plane? 68. three-wheeled car is driving slowly over uneven ground. Is it possible that only two wheels will be touching the ground at a given time? 69. Is it possible to draw three points that do not lie on a plane? 1.3 oints, ines, and lanes 19

7 tandardized est ractice 70. ultiple hoice Which of the statements is false?,, and are collinear.,,, and are coplanar. lies on ^&(. &*( contains &*. ixed eview escribing Number atterns redict the next number. (esson 1.1) 71. 6, 17, 28, 39, , 4, 1, 6, , 20, 100, 500, , 5, 15, 30, 50,... lgebra kills ractions Write the fraction as a decimal. or repeating decimals, round to the nearest hundredth. (kills eview, p. 657) Quiz 1 ketch the next figure you expect in the pattern. (esson 1.1) ind a counterexample to prove that the statement is false. (esson 1.2) 3. If a number is divisible by 10, then it is divisible by wo sides of a triangle can never have the same length. 5. he sum of two numbers is always greater than either number. 6. If you fold a square piece of paper in half, then unfold it and cut along the fold, you will always create two rectangles of the same size. ketch the figure. (esson 1.3) 7. raw three noncollinear points, Q, and. ketch Q &*(. dd a point on the ray so that is between Q and. hen sketch ^&(. 8. raw four points, V, X, Y, and Z, no three of which are collinear. ketch VY ^&(, XZ ^&(, and YZ ^&(. 20 hapter 1 asics of eometry