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Section 8.1 Points, Lines, and Planes 379
8.1 Points, Lines, and Planes
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B P
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Essential QuestionEssential 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
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F
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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, floor, 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, floor, 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, floor, and ceiling in your classroom.
Exploring Dynamic Geometry Software
Work with a partner. Use dynamic geometry software to explore geometry. Use the software to find 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 AnswerCommunicate Your Answer
4. How can you use dynamic geometry software to visualize geometric concepts?
UNDERSTANDING
MATHEMATICAL
TERMS
To be proficient in math, you need to understand defi nitions and previously established results. An appropriate tool, such as a software package, can sometimes help.
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 ProgressMonitoring 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.
undefined terms, p. 380 point, p. 380 line, p. 380 plane, p. 380 collinear points, p. 380 coplanar points, p. 380 defined terms, p. 381 line segment, or segment, p. 381 endpoints, p. 381 ray, p. 381 opposite rays, p. 381 intersection, p. 382
Core VocabularyCore Vocabullarry
What You Will LearnWhat 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 Undefined Terms In geometry, the words point , line , and plane are undefined terms. These words do not have formal definitions, but there is agreement about what they mean.
CoreCore ConceptConcept
Undefined 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 floor 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
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S P
V T^ m
R
n
382 Chapter 8 Basics of Geometry
Sketching Intersections Two or more geometric figures intersect when they have one or more points in common. The intersection of the fi gures is the set of points the figures have in common. Some examples of intersections are shown below.
m A n q
The intersection of two different lines is a point.
The intersection of two different planes is a line.
Sketching Intersections of Lines and Planes
a. Sketch a plane and a line that is in the plane. b. Sketch a plane and a line that does not intersect the plane. c. Sketch a plane and a line that intersects the plane at a point.
SOLUTION
a. b. c.
Sketching Intersections of Planes
Sketch two planes that intersect in a line.
SOLUTION
Step 1 Draw a vertical plane. Shade the plane. Step 2 Draw a second plane that is horizontal. Shade this plane a different color. Use dashed lines to show where one plane is hidden. Step 3 Draw the line of intersection.
Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com
4. Sketch two different lines that intersect a plane at the same point.
Use the diagram.
5. Name the intersection of ⃖ PQ ⃗ and line k. 6. Name the intersection of plane A and plane B. 7. Name the intersection of line k and plane A.
Q M P
B
k A
Section 8.1 Points, Lines, and Planes 383
Solving Real-Life Problems
Modeling with Mathematics
The diagram shows a molecule of sulfur hexafluoride, the most potent greenhouse gas in the world. Name two different planes that contain line r.
A
B
D
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C
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E
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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 ProgressMonitoring 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 hexafluoride as an insulator. Leaks in electrical equipment contribute to the release of sulfur hexafluoride into the atmosphere.
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
X D
E
B
A
Y
✗ AD ⃗ and AC ⃗ are opposite rays.
✗ YC — and YE — are opposite rays.
In Exercises 27–34, use the diagram.
C
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J H
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.
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
Q P
N
K L
M
R
386 Chapter 8 Basics of Geometry
47. REASONING Explain why a four-legged chair may rock from side to side even if the floor is level. Would a three-legged chair on the same level floor rock from side to side? Why or why not? 48. THOUGHT PROVOKING You are designing the living room of an apartment. Counting the floor, walls, and ceiling, you want the design to contain at least eight different planes. Draw a diagram of your design. Label each plane in your design. 49. LOOKING FOR STRUCTURE Two coplanar intersecting lines will always intersect at one point. What is the greatest number of intersection points that exist if you draw four coplanar lines? Explain. 50. HOW DO YOU SEE IT? You and your friend walk in opposite directions, forming opposite rays. You were originally on the corner of Apple Avenue and Cherry Court.
Cherry Ct.
Rose Rd.
Apple Ave.
Daisy Dr.
N E S
W
a. Name two possibilities of the road and direction you and your friend may have traveled. b. Your friend claims he went north on Cherry Court, and you went east on Apple Avenue. Make an argument as to why you know this could not have happened.
MATHEMATICAL CONNECTIONS In Exercises 51–54, graph the inequality on a number line. Tell whether the graph is a segment , a ray or rays , a point , or a line****.
51. x ≤ 3 52. − 7 ≤ x ≤ 4 53. x ≥ 5 or x ≤ − 2 54. ∣^ x ∣^ ≤ 0 55. MODELING WITH MATHEMATICS Use the diagram.
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Q N
P
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M
a. Name two points that are collinear with P. b. Name two planes that contain J. c. Name all the points that are in more than one plane.
CRITICAL THINKING In Exercises 56–63, complete the statement with always , sometimes , or never****. Explain your reasoning.
56. A line ____________ has endpoints. 57. A line and a point ____________ intersect. 58. A plane and a point ____________ intersect. 59. Two planes ____________ intersect in a line. 60. Two points ____________ determine a line. 61. Any three points ____________ determine a plane. 62. Any three points not on the same line ____________ determine a plane. 63. Two lines that are not parallel ___________ intersect. 64. ABSTRACT REASONING Is it possible for three planes to never intersect? intersect in one line? intersect in one point? Sketch the possible situations.
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Find the absolute value. (Skills Review Handbook)
65. ∣ 6 + 2 ∣ 66. ∣ 3 − 9 ∣ 67. ∣ − 8 − 2 ∣ 68. ∣ 7 − 11 ∣
Solve the equation. (Section 1.1)
69. 18 + x = 43 70. 36 + x = 20 71. x − 15 = 7 72. x − 23 = 19
Reviewing what you learned in previous grades and lessons
