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1.1: Points, Lines, and Planes

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Learning Objectives

  • Understand the terms point, line, and plane.
  • Draw and label terms in a diagram.

Review Queue

  1. List and draw pictures of five geometric figures you are familiar with.
  2. What shape is a yield sign?
  3. Solve the algebraic equations.
    1. 4x-7=29
    2. -3x+5=17

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.

How many sides does the octagon have? What is something in-real life that is an octagon?

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. Examples:

Label It Say It
A point A

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

A line has direction and location is always straight.

Label It Say It
line g line g
\overleftrightarrow{\text{PQ}} line PQ
\overleftrightarrow{\text{QP}} line QP

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.

Label It Say It
Plane \mathcal{M} Plane M
Plane ABC Plane ABC

Example 1: What best describes San Diego, California on a globe?

A. point

B. line

C. plane

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

Example 2: What best describes the surface of a movie screen?

A. point

B. line

C. plane

Solution: The surface of a movie screen is most like a plane.

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 goes up and down, and side to side. If we add a third direction, we have space, something three-dimensional.

Collinear: Points that lie on the same line.

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.

Lines h and i and points A, B, C, D, G, and K are coplanar in Plane \mathcal{J}.

Line \overleftrightarrow{KF} and point E are non-coplanar with Plane \mathcal{J}.

Example 3: Use the picture above to answer these questions.

a) List another way to label Plane \mathcal{J}.

b) List another way to label line h.

c) Are K and F collinear?

d) Are E, B and F coplanar?

Solution:

a) Plane BDG. Any combination of three coplanar points that are not collinear would be correct.

b) \overleftrightarrow{AB}. Any combination of two of the letters A, C or B would also work.

c) Yes

d) Yes

Endpoint: A point at the end of a line.

Line Segment: A line with two endpoints. Or, a line that stops at both ends.

Line segments are labeled by their endpoints. Order does not matter.

Label It Say It
\overline{AB} Segment AB
\overline{BA} Segment BA

Ray: 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. For rays, order matters. When labeling, put endpoint under the side WITHOUT an arrow.

Label It Say It
\overrightarrow{CD} Ray CD
\overleftarrow{DC} Ray CD

Intersection: A point or line where lines, planes, segments or rays cross.

Example 4: What best describes a straight road connecting two cities?

A. ray

B. line

C. segment

D. plane

Solution: The straight road connects two cities, which are like endpoints. The best term is segment, or C.

Example 5: Answer the following questions about the picture to the right.

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

b) Are R and Q collinear?

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

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

Solution:

a) No.

b) Yes.

c) S

d) Any combination of P, O, T and Q would work.

Further Beyond This section introduces a few basic postulates.

Postulates: Basic rules of geometry. We can assume that all postulates are true.

Theorem: A statement that is proven true using postulates, definitions, and previously proven theorems.

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

Investigation 1-1: Line Investigation

  1. Draw two points anywhere on a piece of paper.
  2. Use a ruler to connect these two points.
  3. How many lines can you draw to go through these two points?

Postulate 1-2: One plane contains any three non-collinear points.

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

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

Lines l and m intersect at point A.

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

When making geometric drawings, you need to be clear and label all points and lines.

Example 6a: Draw and label the intersection of line \overleftrightarrow{AB} and ray \overrightarrow{CD} at point C.

Solution: It does not matter where you put A or B on the line, nor the direction that \overrightarrow{CD} points.

Example 6b: Redraw Example 6a, so that it looks different but is still true.

Solution:

Example 7: Describe the picture below using the geometric terms you have learned.

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

Know What? Revisited The octagon has 8 sides. In Latin, “octo” or “octa” means 8, so octagon, literally means “8-sided figure.” An octagon in real-life would be a stop sign.

Review Questions

  • Questions 1-5 are similar to Examples 6a and 6b.
  • Questions 6-8 are similar to Examples 3 and 5.
  • Questions 9-12 are similar to Examples 1, 2, and 4.
  • Questions 13-16 are similar to Example 7.
  • Questions 17-25 use the definitions and postulates learned in this lesson.

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

  1. \overrightarrow{CD} intersecting \overline{AB} and Plane P containing \overline{AB} but not \overrightarrow{CD}.
  2. Three collinear points A, B, and C and B is also collinear with points D and E.
  3. \overrightarrow{XY}, \overrightarrow{XZ}, and \overrightarrow{XW} such that \overrightarrow{XY} and \overrightarrow{XZ} are coplanar, but \overrightarrow{XW} is not.
  4. Two intersecting planes, \mathcal{P} and \mathcal{Q}, with \overline{GH} where G is in plane \mathcal{P} and H is in plane \mathcal{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.
  7. Name the geometric figure in three different ways.
  8. Name the geometric figure below in two different ways.
  9. What is the best possible geometric model for a soccer field? Explain your answer.
  10. List two examples of where you see rays in real life.
  11. What type of geometric object is the intersection of a line and a plane? Draw your answer.
  12. What is the difference between a postulate and a theorem?

For 13-16, use geometric notation to explain each picture in as much detail as possible.

For 17-25, determine if the following statements are true or false.

  1. Any two points are collinear.
  2. Any three points determine a plane.
  3. A line is 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 points are coplanar.
  8. \overrightarrow{AB} could be read “ray AB” or “ray “BA.”
  9. \overleftrightarrow{AB} could be read “line AB” or “line BA.”

Review Queue Answers

  1. Examples could be triangles, squares, rectangles, lines, circles, points, pentagons, stop signs (octagons), boxes (prisms), or dice (cubes).
  2. A yield sign is a triangle with equal sides.
    1. 4x-7 = 29\!\\{\;}\quad \ 4x = 36\!\\{\;}\quad \ \ x = 9
    2. -3x+5 =17\!\\{\;}\quad \ -3x = 12\!\\{\;}\qquad \ \ x = -4

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