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1.1: Points, Lines, and Planes

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Learning Objectives

• Understand and build upon the terms:point, line, and plane.
• Apply and use basic postulates.
• Draw and label terms in a diagram.

Review Queue

This subsection will provide a review of Algebra and previous lessons.

1. List and draw pictures of five geometric figures you are familiar with.
2. A plane is an infinitely large, flat, two-dimensional surface. List three examples of objects from real life that resemble planes.
3. Solve the algebraic equations.
1. $4x - 7 =29$
2. $2(-3x + 5) - 8 = -x + 17$
3. $x^2 - 2x - 15 = 0$
4. $x^2 = 121$

Know What? Geometry is everywhere. Remember these wooden blocks that you played with as a kid? If you played with these blocks, then you have been “studying” geometry since you were a child. For example, if you were to move the four triangles at the center of the picture, so that the vertex in the middle was on the outside, what shape would you make? (Flip each triangle outward)

Geometry: The study of shapes and their spatial properties.

Building Blocks

Point: An exact location in space.

A point describes a location, but has no size. Dots are used to represent points in pictures and diagrams.

These points are said “Point $A$,” “Point $L$”, and “Point $F$.” Points are labeled with a CAPITAL letter.

Line: Infinitely many points that extend forever in both directions.

A line, like a point, does not take up space. It has direction, location and is always straight. Lines are one-dimensional because they only have length (no width). A line can by named or identified using any two points on that line or with a lower-case, italicized letter.

This line can be labeled $\overleftrightarrow{P Q}, \ \overleftrightarrow{Q P}$ or just $g$. You would say “line $PQ$,” “line $QP$,” or “line $g$,” respectively. Notice that the line over the $\overleftrightarrow{P Q}$ and $\overleftrightarrow{Q P}$ has arrows over both the $P$ and $Q$. The order of $P$ and $Q$ does not matter.

Plane: Infinitely many intersecting lines that extend forever in all directions.

Think of a plane as a huge sheet of paper that goes on forever. Planes are considered to be two-dimensional because they have a length and a width. A plane can be classified by any three points in the plane.

This plane would be labeled Plane $ABC$ or Plane $\mathcal{M}$. Again, the order of the letters does not matter. Sometimes, planes can also be labeled by a capital cursive letter. Typically, the cursive letter written in a corner of the plane.

Example 1: Which term best describes how San Diego, California, would be represented on a globe?

A. point

B. line

C. plane

Solution: A city is usually labeled with a dot, or point, on a globe. A.

Example 2: Which geometric object best models the surface of a movie screen?

A. point

B. line

C. plane

Solution: The surface of a movie screen extends in two dimensions: up and down and left to right. This description most closely resembles a plane, C.

Beyond the Basics

Now we can use point, line, and plane to define new terms.

Space: The set of all points expanding in three dimensions.

Think back to the plane. It extended along two different lines: up and down, and side to side. If we add a third direction, we have something that looks like three-dimensional space, or the real-world.

Collinear: Points that lie on the same line.

Example 3: Which points are collinear?

Solution: $P, Q, R, S,$ and $T$ are collinear because they are all on line $w$. If a point $U$ was above or below line $w$, it would be non-collinear.

Coplanar: Points and/or lines within the same plane.

Example 4:

a) List two other ways to label Plane $\mathcal{J}$.

b) List one other way to label line $h$.

c) Are $K$ and $F$ collinear? Are they coplanar?

d) Are $E, B$ and $F$ coplanar?

e) List four points that are non-collinear.

Solution:

a) Plane $BDG$, Plane $KAG$, among several others. Any combination of three coplanar points that are not collinear would be correct.

b) $\overleftrightarrow{A B}$ or any combination of two of the letters $A, C$ or $B$ in any order.

c) Yes, they lie on the same line.Yes, you need three points to create a plane, so any two or three points are coplanar.

d) Yes, any three points are coplanar.

e) $A, C, D$ and $E$ would be non-collinear. Since three points define a plane, any three of the points would line in the same plane but the fourth must not.

Endpoint: A point at the end of a line.

Line Segment: Part of a line with two endpoints. Or a line that stops at both ends.

Line segments are labeled by their endpoints, $\overline{AB}$ or $\overline{BA}$. Notice that the bar over the endpoints has NO arrows. Order does not matter.

Ray: Part of a line with one endpoint and extends forever in the other direction.

A ray is labeled by its endpoint and one other point on the line.

Of lines, line segments and rays, rays are the only one where order matters. When labeling, always write the endpoint under the side WITHOUT the arrow, $\overrightarrow{C D}$ or $\overleftarrow{D C}$.

Intersection: A point or set of points where lines, planes, segments or rays cross each other.

Example 5: How do the figures below intersect?

Solution: The first three figures intersect at a point, $P, Q$ and $R$, respectively. The fourth figure, two planes, intersect in a line, $l$. And the last figure, three planes, intersect at one point, $S$.

Example 6: Answer the following questions about the picture to the right.

a) How do the two planes intersect?

b) Is line $l$ coplanar with Plane $\mathcal{V}$ or $\mathcal{W}$?

c) Are $R$ and $Q$ collinear?

d) What point is non-coplanar with either plane?

e) List three coplanar points in Plane $\mathcal{W}$.

Solution:

a) In a line.

b) No.

c) Yes.

d) $S$

e) Any combination of $P, O, T$ and $Q$ would be correct.

Further Beyond

With these new definitions, we can make statements and generalizations about these geometric figures. This section introduces a few basic postulates. Throughout this book we will be introducing Postulates and Theorems so it is important that you understand what they are and how they differ.

Postulates: Basic rules of geometry. We can assume that all postulates are true, much like a definition.

Theorem: A statement that can be proven true using postulates, definitions, and other theorems that have already proven.

The only difference between a theorem and postulate is that a postulate is assumed true because it cannot be shown to be false, a theorem must be proven true. We will prove theorems later in this text.

Postulate 1-1: There is exactly one (straight) line through any two points.

Postulate 1-2: There is exactly one plane that contains any three non-collinear points.

Postulate 1-3: A line with points in a plane also lies within that plane.

Postulate 1-4: The intersection of two distinct lines will be one point.

Postulate 1-5: The intersection of two planes is a line.

When making geometric drawings, you need to be sure to be clear and label. For example, if you draw a line, be sure to include arrows at both ends. Make sure you label your points, lines, and planes clearly, and refer to them by name when writing explanations.

Example 7: Draw and label the intersection of line $\overleftrightarrow{A B}$ and ray $\overrightarrow{C D}$ at point $C$.

Solution: It does not matter the placement of $A$ or $B$ along the line nor the direction that $\overrightarrow{C D}$ points.

Example 8: Describe the picture below using all the geometric terms you have learned.

Solution: $\overleftrightarrow{A B}$ and $D$ are coplanar in Plane $\mathcal{P}$, while $\overleftrightarrow{B C}$ and $\overleftrightarrow{A C}$ intersect at point $C$ which is non-coplanar.

Know What? Revisited If you take the triangles and move them so that the point that met at the center of the square was on the outside, you would get the figure at the right. However, you could also argue that this is not a shape because it has a square hole in the center of it. Another shape that can be made from the four triangles is a rectangle. Part of geometry is justifying and explaining reasoning. You could reason that both of these answers are acceptable.

Review Questions

For questions 1-5, draw and label an image to fit the descriptions.

1. $\overrightarrow{C D}$ intersecting $\overline{A B}$ and Plane $P$ containing $\overline{A B}$ but not $\overrightarrow{C D}$.
2. Three collinear points $A, B,$ and $C$ such that $B$ is also collinear with points $D$ and $E$.
3. $\overrightarrow{X Y}, \ \overrightarrow{X Z},$ and $\overrightarrow{X W}$ such that $\overrightarrow{X Y}$ and $\overrightarrow{X Z}$ are coplanar, but $\overrightarrow{X W}$ is not.
4. Two intersecting planes, $\mathcal{P}$ and $Q$, with $\overline{GH}$ where $G$ is in plane $\mathcal{P}$ and $H$ is in plane $Q$.
5. Four non-collinear points, $I, J, K,$ and $L$, with line segments connecting all points to each other.
6. Name this line in five ways.

1. Name the geometric figure below in two different ways.

1. Draw three ways three different planes can (or cannot) intersect.
2. What type of geometric object is made by the intersection of a sphere (a ball) and a plane? Draw your answer.

For 10-13, use geometric notation to explain each picture in as much detail as possible.

For 14-23, determine if the following statements are ALWAYS true, SOMETIMES true, or NEVER true.

1. Any two distinct points are collinear.
2. Any three points determine a plane.
3. A line is composed to two rays with a common endpoint.
4. A line segment is infinitely many points between two endpoints.
5. A point takes up space.
6. A line is one-dimensional.
7. Any four distinct points are coplanar.
8. $\overrightarrow{A B}$ could be read “ray $AB$” or “ray $BA$.”
9. $\overleftrightarrow{A B}$ could be read “line $AB$” or “line $BA$.”
10. Theorems are proven true with postulates.
11. Two of the above statements are “never.” Explain why.
12. Four of the above statements are “sometimes.” Explain why.

For 26-28, describe the following real world objects in geometric terms.

1. The walls of your classroom and the intersections of these walls with each other and the floor or ceiling. What about where two walls and the floor intersect?
2. The spokes of a bicycle wheel. What about their intersection?
3. Cities on a map. What geometric figure would you draw to measure the distance between them?

In Algebra you plotted points on the coordinate plane and graphed lines. For 29-35, use graph paper and follow the steps to make the diagram on the same graph.

1. Plot the point (2, -3) and label it $A$.
2. Plot the point (-4, 3) and label it $B$.
3. Draw the segment $\overline{AB}$.
4. Locate point $C$, the intersection of this line with the $x-$axis.
5. Draw the ray $\overrightarrow{CD}$ with point $D (1, 4)$.

1. Examples could be triangles, squares, rectangles, lines, circles, points, pentagons, stop signs (octagons), boxes (prisms, or dice (cubes).
2. Examples of a plane would be: a desktop, the chalkboard/whiteboard, a piece of paper, a TV screen, window, wall or a door.
1. $4x-7 = 29\!\\{\;}\quad \ 4x = 36\!\\{\;}\quad \ \ x = 9$
2. $2(-3x + 5) - 8 = -x + 17\!\\{\;}\ -6x + 10 - 8 = -x + 17\!\\{\;}\qquad \ -6x + 2 = -x +17\!\\{\;}\qquad \quad \ \ \ \ -5x = 15\!\\{\;}\qquad \qquad \quad \ \ \ x = 3$
3. Factor, $x = 5, \ -3$
4. $x = \pm 11$

Feb 23, 2012

Sep 22, 2014