10.2: Points, Lines, and Planes (2024)

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    10.2: Points, Lines, and Planes (2)

    Figure 10.2 The lower right-hand corner of The School of Athens depicts a figure representing Euclid illustrating to students how to use a compass on a small chalkboard. (credit: modification of work “School of Athens” by Raphael (1483–1520), Vatican Museums/Wikimedia, Public Domain)

    Learning Objectives

    After completing this section, you should be able to:

    1. Identify and describe points, lines, and planes.
    2. Express points and lines using proper notation.
    3. Determine union and intersection of sets.

    In this section, we will begin our exploration of geometry by looking at the basic definitions as defined by Euclid. These definitions form the foundation of the geometric theories that are applied in everyday life.

    In The Elements, Euclid summarized the geometric principles discovered earlier and created an axiomatic system, a system composed of postulates. A postulate is another term for axiom, which is a statement that is accepted as truth without the need for proof or verification. There were no formal geometric definitions before Euclid, and when terms could not be defined, they could be described. In order to write his postulates, Euclid had to describe the terms he needed and he called the descriptions “definitions.” Ultimately, we will work with theorems, which are statements that have been proved and can be proved.

    Points and Lines

    The first definition Euclid wrote was that of a point. He defined a point as “that which has no part.” It was later expanded to “an indivisible location which has no width, length, or breadth.” Here are the first two of the five postulates, as they are applicable to this first topic:

    1. Postulate 1: A straight line segment can be drawn joining any two points.
    2. Postulate 2: Any straight line segment can be extended indefinitely in a straight line.

    Before we go further, we will define some of the symbols used in geometry in Figure 10.3:

    10.2: Points, Lines, and Planes (3)

    Figure 10.3 Basic Geometric Symbols for Points and Lines

    From Figure 10.3, we see the variations in lines, such as line segments, rays, or half-lines. What is consistent is that two collinear points (points that lie on the same line) are required to form a line. Notice that a line segment is defined by its two endpoints showing that there is a definite beginning and end to a line segment. A ray is defined by two points on the line; the first point is where the ray begins, and the second point gives the line direction. A half-line is defined by two points, one where the line starts and the other to give direction, but an open circle at the starting point indicates that the starting point is not part of the half-line. A regular line is defined by any two points on the line and extends infinitely in both directions. Regular lines are typically drawn with arrows on each end.

    Example 10.1

    Defining Lines

    For the following exercises, use this line (Figure 10.4).

    10.2: Points, Lines, and Planes (4)

    Figure 10.4

    1. Define DE¯DE¯.
    2. Define FF.
    3. Define DFDF.
    4. Define EF¯EF¯.
    Answer
    1. The symbol DE¯DE¯, two letters with a straight line above, refers to the line segment that starts at point DD and ends at point EE.
    2. The letter FF alone refers to point FF.
    3. The symbol DFDF, two letters with a line above containing arrows on both ends, refers to the line that extends infinitely in both directions and contains the points DD and FF.
    4. The symbol EF¯EF¯, two letters with a straight line above, refers to the line segment that starts at point EE and ends at point FF.

    Your Turn 10.1

    For the following exercises, use this line.

    10.2: Points, Lines, and Planes (5)

    Figure 10.5

    1.

    Define /**/\overrightarrow {BD}/**/

    2.

    Define /**/\overline {AB}/**/

    3.

    /**/\overleftarrow {BA}/**/

    4.

    /**/\overleftrightarrow {AD}/**/

    There are numerous applications of line segments in daily life. For example, airlines working out routes between cities, where each city’s airport is a point, and the points are connected by line segments. Another example is a city map. Think about the intersection of roads, such that the center of each intersection is a point, and the points are connected by line segments representing the roads. See Figure 10.6.

    10.2: Points, Lines, and Planes (6)

    Figure 10.6 Air Line Routes

    Example 10.2

    Determining the Best Route

    View the street map (Figure 10.7) as a series of line segments from point to point. For example, we have vertical line segments AB¯AB¯, BC¯,BC¯, and CD¯CD¯ on the right. On the left side of the map, we have vertical line segments HI¯HI¯, FG.¯FG.¯ The horizontal line segments are HA¯HA¯, EI¯EI¯, IE¯IE¯, EB¯EB¯, FC¯FC¯, CF¯,CF¯, and GB¯.GB¯. There are two diagonal line segments, AE¯AE¯ and EF¯.EF¯. Assume that each location is on a corner and that you live next door to the library.

    10.2: Points, Lines, and Planes (7)

    Figure 10.7 Street Map

    1. Let’s say that you want to stop at the grocery store on your way home from school. Come up with three routes you might take to do your errand and then go home. In other words, name the three ways by the line segments in the order you would walk, and which way do you think would be the most efficient route?
    2. How about stopping at the library after school? Name four ways you might travel to the library and which way do you think is the most efficient?
    3. Suppose you need to go to the post office and the dry cleaners on your way home from school. Name three ways you might walk to do your errands and end up at home. Which way do you think is the most efficient way to walk, get your errands done, and go home?
    Answer
    1. From school to the grocery store and home: first way AH¯,HI¯,IE¯,EF¯,FG¯AH¯,HI¯,IE¯,EF¯,FG¯; second way AB¯,BE¯,EI¯,IE¯,EF¯,FG¯AB¯,BE¯,EI¯,IE¯,EF¯,FG¯; third way AE¯,EI¯,IE¯,EF¯,FG¯AE¯,EI¯,IE¯,EF¯,FG¯. It seems that the third way is the most efficient way.
    2. From school to the library: first way AE¯,EF¯,FG¯AE¯,EF¯,FG¯; second way AB¯,BC¯,CD¯,DG¯AB¯,BC¯,CD¯,DG¯; third way AB¯,BE¯,EF¯,FG¯AB¯,BE¯,EF¯,FG¯; fourth way AB¯,BC¯,CF¯,FG¯AB¯,BC¯,CF¯,FG¯. The first way should be the most efficient way.
    3. From school to the post office or dry cleaners to home: first way AE¯,EF¯,FC¯,CD¯,DG¯AE¯,EF¯,FC¯,CD¯,DG¯; second way AB¯,BE¯,EF¯,FC¯,CD¯,DG¯AB¯,BE¯,EF¯,FC¯,CD¯,DG¯; third way AB¯,BC¯,CF¯,FG¯AB¯,BC¯,CF¯,FG¯. The third way would be the most efficient way.

    Your Turn 10.2

    Parallel Lines

    Parallel lines are lines that lie in the same plane and move in the same direction, but never intersect. To indicate that the line l1Figure 10.8.

    10.2: Points, Lines, and Planes (8)

    Figure 10.8 Parallel Lines

    Perpendicular Lines

    Two lines that intersect at a 90Figure 10.9.

    10.2: Points, Lines, and Planes (9)

    Figure 10.9 Perpendicular Lines

    Example 10.3

    Identifying Parallel and Perpendicular Lines

    Identify the sets of parallel and perpendicular lines in Figure 10.10.

    10.2: Points, Lines, and Planes (10)

    Figure 10.10

    Answer

    Drawing these lines on a grid is the best way to distinguish which pairs of lines are parallel and which are perpendicular. Because they are on a grid, we assume all lines are equally spaced across the grid horizontally and vertically. The grid also tells us that the vertical lines are parallel and the horizontal lines are parallel. Additionally, all intersections form a 9090 angle. Therefore, we can safely say the following:

    ABCDABCD, the line containing the points AA and BB is parallel to the line containing the points CC and DD.

    EFGHEFGH, the line containing the points EE and FF is parallel to the line containing the points GG and HH.

    ABEFABEF, the line containing the points AA and BB is perpendicular to the line containing the points EE and FF. We know this because both lines trace grid lines, and intersecting grid lines are perpendicular.

    We can also state that ABGHABGH; the line containing the points AA and BB is perpendicular to the line containing the points GG and HH because both lines trace grid lines, which are perpendicular by definition.

    We also have CDEFCDEF; the line containing the points CC and DD is perpendicular to the line containing the points EE and FF because both lines trace grid lines, which are perpendicular by definition.

    Finally, we see that CDGHCDGH; the line containing the points CC and DD is perpendicular to the line containing the points GG and HH because both lines trace grid lines, which are perpendicular by definition.

    Your Turn 10.3

    1.

    Identify the sets of parallel and perpendicular lines in the given figure.

    10.2: Points, Lines, and Planes (11)

    Figure 10.11

    Defining Union and Intersection of Sets

    Union and intersection of sets is a topic from set theory that is often associated with points and lines. So, it seems appropriate to introduce a mini-version of set theory here. First, a set is a collection of objects joined by some common criteria. We usually name sets with capital letters. For example, the set of odd integers between 0 and 10 looks like this: A={1,3,5,7,9}.A={1,3,5,7,9}. When it involves sets of lines, line segments, or points, we are usually referring to the union or intersection of set.

    The union of two or more sets contains all the elements in either one of the sets or elements in all the sets referenced, and is written by placing this symbol in between each of the sets. For example, let set A={1,2,3},A={1,2,3}, and let set B={4,5,6}.B={4,5,6}. Then, the union of sets A and B is AB={1,2,3,4,5,6}.AB={1,2,3,4,5,6}.

    The intersection of two or more sets contains only the elements that are common to each set, and we place this symbol in between each of the sets referenced. For example, let’s say that set A={1,3,5},A={1,3,5}, and let set B={5,7,9}.B={5,7,9}. Then, the intersection of sets AA and BB is AB={5}.AB={5}.

    Example 10.4

    Defining Union and Intersection of Sets

    Use the line (Figure 10.12) for the following exercises. Draw each answer over the main drawing.

    10.2: Points, Lines, and Planes (12)

    Figure 10.12

    1. Find BDCABDCA.
    2. Find AB¯AD¯AB¯AD¯.
    3. Find ADBC¯ADBC¯.
    4. Find ADBC¯ADBC¯.
    5. Find BACD¯BACD¯.
    Answer
    1. Find BDCABDCA. This is the intersection of the ray BDBD and the ray CA.CA. Intersection includes only the elements that are common to both lines. For this intersection, only the line segment BC¯BC¯ is common to both rays. Thus, BDCA=BC¯.BDCA=BC¯.
      10.2: Points, Lines, and Planes (13)

      Figure 10.13

    2. Find AB¯AD¯AB¯AD¯. The problem is asking for the union of two line segments, AB¯AB¯ and AD¯.AD¯. Union includes all elements in the first line and all elements in the second line. Since AB¯AB¯ is part of AD¯AD¯, AB¯AD¯=AD¯.AB¯AD¯=AD¯.
      10.2: Points, Lines, and Planes (14)

      Figure 10.14

    3. Find ADBC¯ADBC¯. This is the union of the line ADAD with the line segment BC¯BC¯. As the line segment BC¯BC¯ is included on the line AD,AD, then the union of these two lines equals the line ADAD.
      10.2: Points, Lines, and Planes (15)

      Figure 10.15

    4. Find ADBC¯ADBC¯. The intersection of the line ADAD with line segment BC¯BC¯ is the set of elements common to both lines. In this case, the only element in common is the line segment BC¯.BC¯.
      10.2: Points, Lines, and Planes (16)

      Figure 10.16

    5. Find ABCD¯Figure 10.17. Therefore, ABCD¯=ABCD¯=.
      10.2: Points, Lines, and Planes (17)

      Figure 10.17

    Your Turn 10.4

    For the following exercises, use the line shown to identify and draw the union or intersection of sets.

    10.2: Points, Lines, and Planes (18)

    Figure 10.18

    1.

    /**/{\overline {AB}} \cap {\overline {BC}}/**/

    2.

    /**/\overrightarrow {BC} \cup \overleftarrow {CA} /**/

    3.

    /**/{\overline {BC}} \cap {\overline {AC}} /**/

    Planes

    A plane, as defined by Euclid, is a “surface which lies evenly with the straight lines on itself.” A plane is a two-dimensional surface with infinite length and width, and no thickness. We also identify a plane by three noncollinear points, or points that do not lie on the same line. Think of a piece of paper, but one that has infinite length, infinite width, and no thickness. However, not all planes must extend infinitely. Sometimes a plane has a limited area.

    We usually label planes with a single capital letter, such as Plane PFigure 10.19, or by all points that determine the edges of a plane. In the following figure, Plane PP contains points AA and BB, which are on the same line, and point CC, which is not on that line. By definition, PP is a plane. We can move laterally in any direction on a plane.

    10.2: Points, Lines, and Planes (19)

    Figure 10.19 Plane PP

    One way to think of a plane is the Cartesian coordinate system with the xFigure 10.20.

    10.2: Points, Lines, and Planes (20)

    Figure 10.20 Cartesian Coordinate Plane

    This plane contains points SS, TT, and RR. Points TT and RR are colinear and form a line segment. Point SS is not on that line segment. Therefore, this represents a plane.

    To give the location of a point on the Cartesian plane, remember that the first number in the ordered pair is the horizontal move and the second number is the vertical move. Point RR is located at (4,2);(4,2); point SS is located at (3,4);(3,4); and point TT is located at (1,1).(1,1). We can also identify the line segment as TR¯.TR¯.

    Two other concepts to note: Parallel planes do not intersect and the intersection of two planes is a straight line. The equation of that line of intersection is left to a study of three-dimensional space. See Figure 10.21.

    10.2: Points, Lines, and Planes (21)

    Figure 10.21 Parallel and Intersecting Planes

    To summarize, some of the properties of planes include:

    • Three points including at least one noncollinear point determine a plane.
    • A line and a point not on the line determine a plane.
    • The intersection of two distinct planes is a straight line.

    Example 10.5

    Identifying a Plane

    For the following exercises, refer to Figure 10.22

    10.2: Points, Lines, and Planes (22)

    Figure 10.22

    1. Identify the location of points AA, BB, CC, and DD.
    2. Describe the line from point AA to point DD.
    3. Describe the line from point CC containing point BB.
    4. Does this figure represent a plane?
    Answer
    1. Point AA is located at (4,3);(4,3); point BB is located at (4,1);(4,1); point CC is located at (2,3);(2,3); point DD is located at (1,1).(1,1).
    2. The line from point AA to point DD is a line segment AD.¯AD.¯
    3. The line from point CC containing point BB is a ray CBCB starting at point CC in the direction of BB.
    4. Yes, this figure represents a plane because it contains at least three points, points AA and DD form a line segment, and neither point BB nor point CC is on that line segment.

    Your Turn 10.5

    For the following exercises, refer to the given figure.

    10.2: Points, Lines, and Planes (23)

    Figure 10.23

    1.

    Identify the location of points /**/A/**/, /**/B/**/, /**/C/**/, /**/D/**/, /**/E/**/, and /**/F/**/.

    2.

    Describe the line that includes point /**/A/**/ and point /**/B/**/.

    3.

    Describe the line from /**/E/**/ to point /**/F/**/.

    4.

    Describe the line from /**/C/**/ to point /**/D/**/.

    5.

    Does this figure represent a plane?

    Example 10.6

    Intersecting Planes

    Name two pairs of intersecting planes on the shower enclosure illustration (Figure 10.24).

    10.2: Points, Lines, and Planes (24)

    Figure 10.24

    Answer

    The plane ABCDABCD intersects plane CDEFCDEF, and plane CDEFCDEF intersects plane EFGHEFGH.

    Your Turn 10.6

    1.

    Name two pairs of intersecting planes on the shower enclosure shown.

    10.2: Points, Lines, and Planes (25)

    Figure 10.25

    People in Mathematics

    Plato

    Part of a remarkable chain of Greek mathematicians, Plato (427–347 BC) is known as the teacher. He was responsible for shaping the development of Western thought perhaps more powerfully than anyone of his time. One of his greatest achievements was the founding of the Academy in Athens where he emphasized the study of geometry. Geometry was considered by the Greeks to be the “ultimate human endeavor.” Above the doorway to the Academy, an inscription read, “Let no one ignorant of geometry enter here.”

    The curriculum of the Academy was a 15-year program. The students studied the exact sciences for the first 10 years. Plato believed that this was the necessary foundation for preparing students’ minds to study relationships that require abstract thinking. The next 5 years were devoted to the study of the “dialectic.” The dialectic is the art of question and answer. In Plato’s view, this skill was critical to the investigation and demonstration of innate mathematical truths. By training young students how to prove propositions and test hypotheses, he created a culture in which the systematic process was guaranteed. The Academy was essentially the world’s first university and held the reputation as the ultimate center of learning for more than 900 years.

    Check Your Understanding

    Use the graph for the following exercises.
    10.2: Points, Lines, and Planes (26)

    1.

    Identify the type of line containing point /**/D/**/ and point /**/A/**/.

    2.

    Identify the type of line containing points /**/C/**/ and /**/B/**/.

    3.

    Identify the type of line that references point /**/E/**/ and contains point /**/F/**/.

    Use the line for the following exercises.

    10.2: Points, Lines, and Planes (27)

    4.

    Determine /**/{\overline {AB}} \cup {\overline {BD}}/**/.

    5.

    Determine /**/\overleftarrow {BD} \cap {\overline {BC}} /**/.

    6.

    Determine /**/\overleftarrow {BA} \cup \overrightarrow {BD} /**/.

    7.

    How can you determine whether two lines are parallel?

    8.

    How can you determine whether two lines are perpendicular?

    9.

    Determine if the illustration represents a plane.
    10.2: Points, Lines, and Planes (28)

    Section 10.1 Exercises

    Use this graph for the following exercises.
    10.2: Points, Lines, and Planes (29)

    1.

    Identify the kind of line passing through the points /**/A/**/ and /**/B/**/.

    2.

    Identify the kind of line passing through the points /**/C/**/ and /**/D/**/.

    3.

    Identify all sets of parallel lines.

    4.

    Identify all sets of perpendicular lines.

    Use the line for the following exercises.

    10.2: Points, Lines, and Planes (30)

    5.

    Find /**/\overleftarrow {BA} \cup {\overline {BC}} /**/.

    6.

    Find /**/\overrightarrow {AD} \cup \overrightarrow {CD} /**/.

    7.

    Find /**/{\overline {AC}} \cap {\overline {BD}} /**/.

    8.

    Find /**/\overleftarrow {CA} \cap \overrightarrow {BD} /**/

    9.

    Find /**/{\overline {AB}} \cup {\overline {BD}} /**/

    10.

    Find /**/\overleftrightarrow {BC} \cup {\overline {BD}} /**/

    For the following exercises, reference this line.

    10.2: Points, Lines, and Planes (31)

    11.

    Name the points in the set /**/\overleftrightarrow {FH} \cap {\overline {GH}}/**/.

    12.

    Name the points contained in the set /**/\overrightarrow {HL} \cap {\overline {JL}}/**/.

    13.

    Name the points in the set /**/\overleftrightarrow {EL} \cup {\overline {GH}}/**/.

    14.

    Name the points in the set /**/\overleftarrow {GE} \cup \overrightarrow {KL} /**/.

    15.

    Name the points in the set /**/\overleftarrow {GE} \cap \overrightarrow {KL} /**/.

    16.

    Name the points in the set /**/\overrightarrow {HK} \cap \overleftarrow {JF} /**/.

    17.

    Name the points in the set /**/\overrightarrow {GL} \cap K/**/.

    18.

    Name the points in the set /**/\overleftarrow {HE} \cup \overrightarrow {KL} /**/.

    Use the given figure for the following exercises.
    10.2: Points, Lines, and Planes (32)

    19.

    Describe the lighter plane (right) in the drawing.

    20.

    Describe the darker plane (left) in the drawing.

    10.2: Points, Lines, and Planes (2024)
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