Points, Lines, and Planes 8.1, Assignments of Geometry

Download Points, Lines, and Planes 8.1 and more Assignments Geometry in PDF only on Docsity! Section 8.1 Points, Lines, and Planes 379 Points, Lines, and Planes8.1 Q PB A Essential Question How can you use dynamic geometry software to visualize geometric concepts? Using Dynamic Geometry Software Work with a partner. Use dynamic geometry software to draw several points. Also, draw some lines, line segments, and rays. What is the difference between a line, a line segment, and a ray? Sample A B C F G D E Intersections of Lines and Planes Work with a partner. a. Describe and sketch the ways in which two lines can intersect or not intersect. Give examples of each using the lines formed by the walls, fl oor, and ceiling in your classroom. b. Describe and sketch the ways in which a line and a plane can intersect or not intersect. Give examples of each using the walls, fl oor, and ceiling in your classroom. c. Describe and sketch the ways in which two planes can intersect or not intersect. Give examples of each using the walls, fl oor, and ceiling in your classroom. Exploring Dynamic Geometry Software Work with a partner. Use dynamic geometry software to explore geometry. Use the software to fi nd a term or concept that is unfamiliar to you. Then use the capabilities of the software to determine the meaning of the term or concept. Communicate Your Answer 4. How can you use dynamic geometry software to visualize geometric concepts? UNDERSTANDING MATHEMATICAL TERMS To be profi cient in math, you need to understand defi nitions and previously established results. An appropriate tool, such as a software package, can sometimes help. int_math1_pe_0801.indd 379 1/29/15 3:40 PM 380 Chapter 8 Basics of Geometry 8.1 Lesson Collinear points are points that lie on the same line. Coplanar points are points that lie in the same plane. Naming Points, Lines, and Planes a. Give two other names for ⃖ ⃗ PQ and plane R. b. Name three points that are collinear. Name four points that are coplanar. SOLUTION a. Other names for ⃖ ⃗ PQ are ⃖ ⃗ QP and line n. Other names for plane R are plane SVT and plane PTV. b. Points S, P, and T lie on the same line, so they are collinear. Points S, P, T, and V lie in the same plane, so they are coplanar. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. Use the diagram in Example 1. Give two other names for ⃖ ⃗ ST . Name a point that is not coplanar with points Q, S, and T. undefi ned terms, p. 380 point, p. 380 line, p. 380 plane, p. 380 collinear points, p. 380 coplanar points, p. 380 defi ned terms, p. 381 line segment, or segment, p. 381 endpoints, p. 381 ray, p. 381 opposite rays, p. 381 intersection, p. 382 Core Vocabulary What You Will Learn Name points, lines, and planes. Name segments and rays. Sketch intersections of lines and planes. Solve real-life problems involving lines and planes. Using Undefi ned Terms In geometry, the words point, line, and plane are undefi ned terms. These words do not have formal defi nitions, but there is agreement about what they mean. Core Concept Undefi ned Terms: Point, Line, and Plane Point A point has no dimension. A dot represents a point. Line A line has one dimension. It is represented by a line with two arrowheads, but it extends without end. Through any two points, there is exactly one line. You can use any two points on a line to name it. Plane A plane has two dimensions. It is represented by a shape that looks like a fl oor or a wall, but it extends without end. Through any three points not on the same line, there is exactly one plane. You can use three points that are not all on the same line to name a plane. A point A A B line , line AB (AB), or line BA (BA) A C M B plane M, or plane ABC Q PS T mV R n int_math1_pe_0801.indd 380 1/29/15 3:40 PM Section 8.1 Points, Lines, and Planes 383 Solving Real-Life Problems Modeling with Mathematics The diagram shows a molecule of sulfur hexafl uoride, the most potent greenhouse gas in the world. Name two different planes that contain line r. A B D G C F E q p r SOLUTION 1. Understand the Problem In the diagram, you are given three lines, p, q, and r, that intersect at point B. You need to name two different planes that contain line r. 2. Make a Plan The planes should contain two points on line r and one point not on line r. 3. Solve the Problem Points D and F are on line r. Point E does not lie on line r. So, plane DEF contains line r. Another point that does not lie on line r is C. So, plane CDF contains line r. Note that you cannot form a plane through points D, B, and F. By defi nition, three points that do not lie on the same line form a plane. Points D, B, and F are collinear, so they do not form a plane. 4. Look Back The question asks for two different planes. You need to check whether plane DEF and plane CDF are two unique planes or the same plane named differently. Because point C does not lie on plane DEF, plane DEF and plane CDF are different planes. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use the diagram that shows a molecule of phosphorus pentachloride. G K L I J H s 8. Name two different planes that contain line s. 9. Name three different planes that contain point K. 10. Name two different planes that contain ⃗ HJ. Electric utilities use sulfur hexafl uoride as an insulator. Leaks in electrical equipment contribute to the release of sulfur hexafl uoride into the atmosphere. int_math1_pe_0801.indd 383 1/29/15 3:40 PM 384 Chapter 8 Basics of Geometry Exercises8.1 Dynamic Solutions available at BigIdeasMath.com 1. WRITING Compare collinear points and coplanar points. 2. WHICH ONE DOESN’T BELONG? Which term does not belong with the other three? Explain your reasoning. — AB plane CDE ⃖ ⃗ FG ⃗ HI Vocabulary and Core Concept Check In Exercises 3–6, use the diagram. A C EDT SB 3. Name four points. 4. Name two lines. 5. Name the plane that contains points A, B, and C. 6. Name the plane that contains points A, D, and E. In Exercises 7–10, use the diagram. (See Example 1.) T S f R V Q W g 7. Give two other names for ⃖ ⃗ WQ. 8. Give another name for plane V. 9. Name three points that are collinear. Then name a fourth point that is not collinear with these three points. 10. Name a point that is not coplanar with R, S, and T. In Exercises 11–16, use the diagram. (See Example 2.) B C D A E t s 11. What is another name for — BD ? 12. What is another name for — AC ? 13. What is another name for ray ⃗ AE? 14. Name all rays with endpoint E. 15. Name two pairs of opposite rays. 16. Name one pair of rays that are not opposite rays. In Exercises 17–24, sketch the fi gure described. (See Examples 3 and 4.) 17. plane P and line ℓ intersecting at one point 18. plane K and line m intersecting at all points on line m 19. ⃗ AB and ⃖ ⃗ AC 20. ⃗ MN and ⃗ NX 21. plane M and ⃗ NB intersecting at B 22. plane M and ⃗ NB intersecting at A 23. plane A and plane B not intersecting 24. plane C and plane D intersecting at ⃖ ⃗ XY Monitoring Progress and Modeling with Mathematics int_math1_pe_0801.indd 384 1/29/15 3:40 PM Section 8.1 Points, Lines, and Planes 385 ERROR ANALYSIS In Exercises 25 and 26, describe and correct the error in naming opposite rays in the diagram. C DX E B A Y 25. ⃗ AD and ⃗ AC are opposite rays.✗ 26. — YC and — YE are opposite rays.✗ In Exercises 27–34, use the diagram. C G HJ I D E A B F 27. Name a point that is collinear with points E and H. 28. Name a point that is collinear with points B and I. 29. Name a point that is not collinear with points E and H. 30. Name a point that is not collinear with points B and I. 31. Name a point that is coplanar with points D, A, and B. 32. Name a point that is coplanar with points C, G, and F. 33. Name the intersection of plane AEH and plane FBE. 34. Name the intersection of plane BGF and plane HDG. In Exercises 35–38, name the geometric term modeled by the object. 35. 36. 37. 38. In Exercises 39–44, use the diagram to name all the points that are not coplanar with the given points. 39. N, K, and L 40. P, Q, and N 41. P, Q, and R 42. R, K, and N 43. P, S, and K 44. Q, K, and L 45. CRITICAL THINKING Given two points on a line and a third point not on the line, is it possible to draw a plane that includes the line and the third point? Explain your reasoning. 46. CRITICAL THINKING Is it possible for one point to be in two different planes? Explain your reasoning. S PQ N LK M R int_math1_pe_0801.indd 385 1/29/15 3:40 PM