1-3 Points, Lines, Planes 5e3l6n

1-3

1-3

Points, Lines, and Planes

1. Plan TEKS

GO for Help

Check Skills You’ll Need (G.1)(A) To develop an awareness of the structure of a mathematical system, connecting definitions and postulates (G.3)(D) To use inductive reasoning to formulate a conjecture

(G.1)(A) To develop an awareness of the structure of a mathematical system, connecting definitions and postulates

(G.3)(D) To use inductive reasoning to formulate a conjecture

Dana Center TEKS Toolkit Clarifying Activities Activities (G.1)(A), (G.3)(D) Visit: PHSchool.com Web Code: auq-9045

Skills Handbook page 760

2

x Algebra Solve each system of equations. 1. y = x + 5 (1, 6) y = -x + 7

2. y = 2x - 4 (3, 2) y = 4x - 10

4. Copy the diagram of the four points A, B, C, and D. Draw as many different lines as you can to connect pairs of points. See margin, p. 17.

3. y = 2x (5, 10) y = -x + 15 A B C D

New Vocabulary • point • space • line • collinear points • coplanar • postulate • axiom

• plane

1

1 1 Basic of Geometry Part TEKS Math Background The formal study of geometry requires simple ideas and statements that can be accepted as true without proof. The undefined point, line, and plane provide the simple ideas. Basic postulates about points, lines and planes can be accepted without proof. These form the building blocks for the first theorems that students can prove.

Hands-On Activity: How Many Lines Can You Draw? Many constellations are named for animals and mythological figures. It takes some imagination to the points representing the stars to get a recognizable figure such as Leo the Lion. How many lines can you draw connecting the 10 points in Leo the Lion? • Make a table and look for a pattern to help you find out.

More Math Background: p. 2C

1. Mark three points on a circle. Now connect the three points with as many (straight) lines as possible. How many lines can you draw? 3 lines

Lesson Planning and Resources

2. Mark four points on another circle. How many lines can you draw to connect the four points? 6 lines 3. Repeat this procedure for five points on a circle and then for six points. How many lines can you draw to connect the points? 10 lines, 15 lines

See p. 2E for a list of the resources that this lesson.

4. Use inductive reasoning to tell how many lines you can draw to connect the ten points of the constellation Leo the Lion. 45 lines PowerPoint

Bell Ringer Practice Check Skills You’ll Need

In geometry, some words such as point, line, and plane are undefined. In order to define these words you need to use words that need further defining. It is important however, to have general descriptions of their meanings.

Use student page, transparency, or PowerPoint.

Daily Review Use transparency 3.

16

Chapter 1 Tools of Geometry

Special Needs

Below Level

L1

For Example 4, pair students with visual difficulties with visual learners to identify and name planes.

16

learning style: visual

L2

Encourage students to use pencils and sheets of paper to model Postulates 1-1, 1-2, and 1-3. Students may cut partially through each sheet and the sheets to model Postulate 1-3.

learning style: tactile

You can think of a point as a location. A point has no size. It is represented by a small dot and is named by a capital letter. A geometric figure is a set of points. Space is defined as the set of all points.

Guided Instruction

t B

Hands-On Activity

A

Vocabulary Tip

* )

You can think of a line as a series of points that extends in two opposite without end. You can name a line by any two points on the line, such as ) *directions AB (read “line AB”). Another way to name a line is with a single lowercase letter, such as line t (see above). Points that lie on the same line are collinear points.

* )

AB (“line AB”) and BA

(“line BA”) name the same line.

2. Teach

Some students may count lines in both “directions.” Point out that each line should be counted only once. PowerPoint

1

EXAMPLE

Additional Examples

Identifying Collinear Points

a. Are points E, F, and C collinear? If so, name the line on which they lie.

1 In the figure below, name three points that are collinear and three points that are not collinear. Y, Z, and W are collinear; X, Y, and Z and X, W, and Z are not collinear.

n E

Points E, F, and C are collinear. They lie on line m.

C

F

D

P

m !

b. Are points E, F, and D collinear? If so, name the line on which they lie.

A

Points E, F, and D are not collinear.

Quick Check

B

1 a. Are points F, P, and C collinear? no 4 4 4 b. Name line m in three other ways. Answers may vary. Sample: EF, FC, CE c. Critical Thinking Why* do) you think arrowheads are used when drawing a line or naming a line such as EF ? Arrowheads are used to show that the line extends in opposite directions without end.

A plane is a flat surface that has no thickness. A plane contains many lines and extends without end in the directions of all its lines. You can name a plane by either a single capital letter or by at least three of its noncollinear points. Points and lines in the same plane are coplanar. B

P A

2

R

U

S

T

A

Naming a Plane

C

You can name the plane represented by the front of the ice cube using at least three noncollinear points in the plane. Some names are plane AEF, plane AEB, and plane ABFE.

H

D

G

E

F

C

D A

B

2 List three different names for the plane represented by the top of the ice cube. Answers may vary. Sample: HEF, HEFG, FGH

Lesson 1-3 Points, Lines, and Planes

Advanced Learners

2 Name the plane shown in two different ways. Sample: plane RST, plane RSTU

B

EXAMPLE

Each surface of the ice cube represents part of a plane. Name the plane represented by the front of the ice cube.

Quick Check

D

page 16 Check Skills You’ll Need 4.

C

Plane ABC

Plane P

C

L4

In Example 4, have students find the number of different planes named by any three vertices of the cube.

17

English Language Learners ELL The word noncollinear in Postulate 1-4 may confuse students. Discuss how the prefix non- indicates not. So, noncollinear points do not lie on a line.

learning style: verbal

17

Guided Instruction

2 1

Basic Postulates of Geometry

Tactile Learners A postulate or axiom is an accepted statement of fact.

Have students use index cards to illustrate Postulates 1-3 and 1-4.

3

EXAMPLE

You have used some of the following geometry postulates in algebra. For example, you used Postulate 1-1 when you graphed an equation such as y = -2x + 8. You plotted two points and then drew the line through those two points.

Math Tip

The drawing of a plane is necessarily finite. Remind students that the intersection is a line, not a segment, and that they should use a line symbol above the letters.

Key Concepts

Postulate 1-1 Through any two points there is exactly one line.

t B

Line t is the only line that es through points A and B.

A

PowerPoint

Additional Examples 3 Use the diagram from Example 3. What is the intersection of plane * ) HGC and plane AED? HD

Vocabulary Tip There is exactly one means “there is one and there is no more than one.”

y y ! 3x # 7

2 (3, 2) y ! #2x $ 8 O

y = 3x - 7 show, the two lines intersect at a single point, (3, 2). The solution to the system of equations is (3, 2).

Z

V

4

y = -2x + 8

4 Shade the plane that contains X, Y, and Z.

Y

In algebra, one way to solve a system of two equations is to graph the two equations. As the graphs of

1

5

3

#4

This illustrates Postulate 1-2.

X

Key Concepts

W

Postulate 1-2 If two lines intersect, then they intersect in exactly one point.

Resources • Daily Notetaking Guide 1-3 L3 • Daily Notetaking Guide 1-3— L1 Adapted Instruction

* )

* )

AE and BD intersect at C.

A

B C E

D

There is a similar postulate about the intersection of planes.

Closure Explain how a postulate and a conjecture are alike and how they are different. Sample: You accept a postulate as true without proof, but you try to determine whether a conjecture is true or false.

Key Concepts

Postulate 1-3 If two planes intersect, then they intersect in exactly one line. Plane RST and * ) plane STW intersect in ST.

R S

T

W

When you know two points in the intersection of two planes, Postulates 1-1 and 1-3 tell you that the line through those points is the line of intersection of the planes.

18

18

x

7

#2

Chapter 1 Tools of Geometry

3

Quick Check

3

3. Practice

Finding the Intersection of Two Planes

EXAMPLE

What is the intersection of plane HGFE and plane BCGF?

E

Plane HGFE * and ) plane BCGF intersect in GF .

A

H

G

Assignment Guide

F

1 A B 1-16, 46-60, 64, 68-73

C

D B

2 A B

* .) Name two planes that intersect in

C Challenge

BF

17-45, 61-63, 65-67 74-79

ABF and CBF

Tune-Up Mixed Review

A three-legged stand will always be stable. As long as the feet of the stand don’t lie in one line, the feet of the three legs will lie exactly in one plane.

Homework Quick Check To check students’ understanding of key skills and concepts, go over Exercises 5, 18, 50, 62, 66.

This illustrates Postulate 1-4.

Key Concepts

Postulate 1-4

Error Prevention!

Through any three noncollinear points there is exactly one plane.

4

Exercises 9, 10 Check that students use a line symbol and not a segment or ray symbol as they name lines.

Using Postulate 1-4

EXAMPLE

a. Shade the plane that contains A, B, and C. H

b. Shade the plane that contains E, H, and C. H

G

E

F

C

D

G

E

F

A

Quick Check

80-82 83-88

C

D A

B

B

GPS Guided Problem Solving

L3 L4

Enrichment

4 a. Name another point that is in the same plane as points A, B, and C. D b. Name another point that is coplanar with points E, H, and C. B

L2

Reteaching

L1

Adapted Practice Practice Name

Class

L3

Date

Practice 1-3

Points, Lines, and Planes

Refer to the diagram at the right for Exercises 1–15.

EXERCISES

* )

1. Name AB in another way.

For more exercises, see Extra Skill, Word Problem, and Proof Practice.

2. Give two other names for plane Q. 3. Why is EBD not an acceptable name for plane Q? A

5. B and F

6. EB and A

7. F and plane Q

Are the following sets of points coplanar?

Example 1

GO for Help

(page 17)

3. yes; line n 5. yes; line n

1. A, D, E no

14. plane Q and EC

15. FB and BD

3. B, C, F 5. F, B, D 7. G, F, C no

A

* )

16. GK and LG

m

H

G

17. planes GLM and LPN L

K

18. planes GHPN and KJP

M

J

* ) * )

G C

N

P

20. KP and plane KJN

6. F, A, E no

21. KM and plane GHL

F

D

Refer to the diagram at the right.

n

22. Name plane P in another way. D

23. Name plane Q in another way. 24. What is the intersection of planes P and Q?

B

25. Are A and C collinear?

C

B Q

A

26. Are D, A, B, and C coplanar? 27. Are D and C collinear?

E

9. Name line m in three other ways.

* )

Find the intersection of the following lines and planes in the figure at the right.

19. planes HJN and GKL

4. A, E, C yes; line m

8. A, G, C yes; line m

* )

C

F

13. F, A, B, and D

* )

B

* ) * )

9. DB and FC

11. AE and DC

12. F, A, B, and C

* )

2. B, C, D yes; line n

* ) * )

8. E, B, and F

10. AC and ED

* )

Are the three points collinear? If so, name the line on which they lie.

D E

4. AB and C

* )

Practice by Example

Q

Are the following sets of points collinear?

* ) * )

Practiceand andProblem ProblemSolving Solving Practice

* )

* )

P

28. What is the intersection of AB and DC ? 29. Are planes P and Q coplanar?

* )

30. Are AB and plane Q coplanar?

9. Answers may vary. 4 4 4 Sample: AE, EC, GA

31. Are B and C collinear?

10. Name line n in three other ways. 4 4 4 Answers may vary. Sample: BF, CD, DF Lesson 1-3 Points, Lines, and Planes

19

19

Exercises 38–43 These exercises

Example 2

introduce the term noncoplanar. Discuss with students how they can derive its meaning from what they already know about the collinear and noncollinear.

(page 17)

Exercises 47–50 Discuss the

Example 3

exercises for which students think that drawing a figure for the description is not possible. Have students explain their reasoning. This is a good way to clarify the ideas that form the basis of a formal proof.

(page 19)

Name the plane represented by each surface of the box. 11. the bottom ABCD

12. the top EFHG

13. the front ABHF

14. the back EDCG

15. the left side EFAD

16. the right side BCGH

G

E

F

X

4 18. planes UXV and WVS VW 4 19. planes XWV and UVR 20. planes TXW and TQU XT 4 UV Name two planes that intersect in the given line.

Example 4

Exercises 56–61 These exercises

(page 19)

reinforce the vocabulary and postulates in the lesson. Have students work with partners to discuss any unclear . Emphasize the mathematical importance of the phrase exactly one in Exercise 57. Also point out in Exercise 61 that two lines means two distinct lines.

* )

*

*

)

W

U

4 17. RS 17. planes QRS and RSW

Error Prevention!

B

A

Use the figure at the right for Exercises 17–37. First, name the intersection of each pair of planes.

* )

H

C

D

V

S

T Q

)

R

21. QU 22. TS 23. XT 24. VW Exercises 17–37 QUX and QUV XTS and QTS UXT and WXT UVW and RVW Copy the figure. Shade the plane that contains the given points. 25. R, V, W 26. U, V, W 27. U, X, S 25–29. See back of book. Name another point in each plane.

28. T, U, X

29. T, V, R

30. plane RVW 31. plane UVW 32. plane UXS 33. plane TUX 34. plane TVR S X R Q X Is the given point coplanar with the other three points?

Apply Your Skills

Careers Exercise 66 Ask: What fact about Earth’s surface complicates the work of a surveyor who uses lines and planes? Earth is a sphere and not flat, so distances are measured along curves.

37. point W with X, V, R 35. point Q with V, W, S 36. point U with T, V, S no yes no Postulate 1-4 states that any three noncollinear points lie in one plane. Find the plane containing the first three points listed, then decide whether the fourth point is in that plane. Z Y Write coplanar or noncoplanar to describe the points. S U 39. S, U, V, Y coplanar 38. Z, S, Y, C coplanar V X 40. X, Y, Z, Unoncoplanar 41. X, S, V, U coplanar C 42. X, Z, S, V noncoplanar 43. S, V, C, Y noncoplanar 44. Photography Photographers and surveyors use a tripod, or three-legged stand, for their instruments. Use one of the postulates to explain why. See margin.

* )

45. Open-Ended Draw a figure with points B, C, D, E, F, and G that shows CD , * ) * ) BG , and EF , with one of the points on all three lines. See margin.

44. Through any three noncollinear points there is exactly one plane. The ends of the legs of the tripod represent three noncollinear points, so they rest in one plane. Therefore, the tripod won’t wobble.

If possible, draw a figure to fit each description. Otherwise write not possible. 46–49. See margin. 46. four points that are collinear 47. two points that are noncollinear 48. three points that are noncollinear Real-World

Connection

Careers The photographer uses a tripod to help assure a clear picture.

49. three points that are noncoplanar

50–53. See back of book. Coordinate Geometry Graph the points and state whether they are collinear.

GPS 50. (3, -3), (2, -3), (-3, 1) 52. (2, -2), (-2, -2), (3, -2)

51. (2, 2), (-2, -2), (3, 2) 53. (-3, 3), (-3, 2), (-3, -1)

54. Obj. 10: (8.14)(B) Which three points are not collinear? C (0, 0), (0, 2), (0, 4) (0, 0), (0, 2), (3, 0) (0, 0), (3, 0), (5, 0) (2, -2), (2, 2), (2, 3)

45. Answers may vary. Sample: E

D

B CF

n

46. A

B

C

D

20

Chapter 1 Tools of Geometry

G

47. not possible

49. not possible

48. A

62.

B

A C

C B

20

N

Post. 1-4: Through three noncollinear points there is exactly one plane.

63. Answers may vary. Sample: M V

P L

N

K

GO

nline

Homework Help Visit: PHSchool.com Web Code: aue-0103

4. Assess & Reteach

Use always, sometimes, or never to make a true statement. 55. Intersecting lines are 9 coplanar. always 56. Two planes 9 intersect in exactly one point. never

PowerPoint

Lesson Quiz

57. Three points are 9 coplanar. always 58. A plane containing two points of a line 9 contains the entire line. always

Q

61. How many planes contain each line and point? * ) * ) a. EF and point G 1 b. PH and point E 1 * ) * ) c. FG and point P 1 d. EP and point G 1 e. Make a Conjecture What do you think is H true of a line and a point not on the line? E A line and a point not on the line are always coplanar.

D

1. Name three collinear points. D, J, and H

F

2. Name two different planes that contain points C and G. planes BCGF and CGHD 3. Name the intersection AED and plane HEG. *of plane ) HE

62. The noncollinear points A, B, and C are all contained in plane N.

)

NM .

4. How many planes contain the points A, F, and H? 1

64. Optical Illusions The diagram (right) is an optical illusion. A Which three points are collinear: A, B, and C or A, B, and D? Are you sure? Use a straightedge to check your answer. A, B, and D B Writing Use postulates to explain each situation. 65–67. See margin.

5. Show that this conjecture is false by finding one counterexample: Two planes always intersect in exactly one line. Sample: Planes AEHD and BFGC never intersect.

65. A land surveyor can always find a straight line from the point where she stands to any other point she can see. 66. A carpenter knows that a line can represent the intersection of two flat walls.

C

D

67. A furniture maker knows that a three-legged table is always steady, but a four-legged table will sometimes wobble. 68–72. See back of book. 73. See margin. Coordinate Geometry Graph the points and state whether they are collinear.

Alternative Assessment

68. (1, 1), (4, 4), (-3, -3)

69. (2, 4), (4, 6), (0, 2)

70. (0, 0), (-5, 1), (6, -2)

71. (0, 0), (8, 10), (4, 6)

72. (0, 0), (0, 3), (0, -10)

73. (-2, -6), (1, -2), (4, 1)

74. How many planes contain the same three collinear points? Explain. See left. 75. Navigation Rescue teams use Postulates 1-1 and 1-2 to determine the location of a distress signal. In the diagram, a ship at point A receives a signal from the northeast. A ship at point B receives the same signal from due west. Trace the diagram and find the location of the distress signal. Explain how the two postulates help locate the distress signal. See margin.

lesson quiz, PHSchool.com, Web Code: aua-0103

65. Post. 1-1: Through any two points there is exactly one line. 66. Post. 1-3: If two planes intersect, then they intersect in exactly one line.

H

J

Use the diagram above. G

In Exercise 62 and 63, sketch a figure for the given information. Then name the postulate that your figure illustrates. 62–63. See margin, pp. 20–21.

74. Infinitely many; explanations may vary. Sample: Infinitely many planes can intersect in one line.

E

A

P

Challenge

G

C

60. Two lines 9 meet in more than one point. never

* 63. Planes LNP and MVK intersect in

F

B

59. Four points are 9 coplanar. sometimes

W

Have students write Postulate 1-4, illustrate it, and explain it in their own words. Then have them explain how changing the word three to four or the word plane to line makes the postulate unreasonable.

B

NE D

75.

B

A Distress Signal A

Lesson 1-3 Points, Lines, and Planes

67. The end of one leg might not be coplanar with the ends of the other three legs. (Post. 1-4)

73.

4

y 2

O 24

no

x

21

By Post. 1-1, points D and B determine a line and points A and D determine a line. The distress signal is on both lines and, by Post. 1-2, there can be only one distress signal.

21

76. a. Open-Ended Suppose two points are in plane P. Explain why it makes sense that the line containing the points would be in the same plane. See margin. b. Suppose two lines intersect. How many planes do you think contain both lines? A C You may use the diagram and your answer in B part (a) to explain your answer. See margin.

• TAKS Tune Up, p. 75 • TAKS Strategies, p. 70 • TAKS Daily Review Transparencies

• TAKS Review and Preparation

Probability Points are picked at random from A, B, C, and D, which are arranged as shown. Find the probability that the indicated number of points meet the given condition.

Workbook • TAKS Strategies with Transparencies

77. 2 points, collinear 1

D A

78. 3 points, collinear 14

B

C

79. 3 points, coplanar 1

Tune-Up

76. a. Since the plane is flat, the line would have to curve so as to contain the 2 points and not lie in the plane; but lines are straight.

Objective 2 TEKS (A.3)(A)

80. The price of a shirt is decreased by 30% after $10 is subtracted from the original price p. Which expression can help you find how much the shirt sells for now? B (p - 0.3p) - 10 0.3(p - 10) p - 10 - 0.3(p - 10)

b. One plane; Points A, B, and C are noncollinear. By Post. 1-4, they are coplanar. Then, by part (a), 4 4 AB and BC are coplanar.

Objective 9 TEKS (8.11)(B)

(p - 10) - 0.3p

81. The results of each toss of a number cube are shown below. Based on the results, what is the experimental probability of rolling a 5? H 1 5 5 6 5 2 4 4 3 5 15%

Objective 9 TEKS (8.12)(C)

25%

40%

50%

82. The data below show the number of minutes Anna spent exercising each day last week. Which box-and-whisker plot best represents the data? 0 10 11 12 12 14 16 17 19 20 C

0

11

13

17

20

0

5

10

15

20

0

5

10

15

20

0

5

10

15

20

Mixed Review

GO for Help

Lesson 1-2 TEKS (G.6)(C)

Make an orthographic drawing for each figure. 83–85. See margin. 83.

84.

t

Fro

h Rig

nt

Skills Handbook

Rig

Fro

ht

nt

Simplify each ratio. 2 87. pr 2r 2pr

88. 5 : 75 13 21

Chapter 1 Tools of Geometry

83.

84.

Front

85. Top

Top

22

Fro

nt

86. 30 to 12 5 to 2

22

85.

Right

Front

Top

Right

Front

Right

ht Rig