1.2  Points, Lines, & Planes

Point:

A point names a location and :

• Has no size
• It can represented by a dot or symbol
• Is at the intersection of two or more lines ( even curves! )
• Is labeled using a capital letter.
  

Example A

Question: Could the following be used to identify a point? P A X M

Solution: Since a point has no size or shape we can use just about anything to identify a point.  Remember it is just a location, a place we need to identify.

Watch the video:

Line:  

A line is

• A straight path
• Has no thickness
• Extends forever, in opposite directions from any given point on the line
  • In other words, a line does not have a starting point or an endpoint
• There are an infinite number of points on a line.
• A line is identified by any two point,
• A line is named using either a lowercase letter or any two points on the line
  • When a line is named by two points, a ( ↔ ) line with arrows on each each end is drawn over the two points

Since a line is made up of an infinite number of points, any set ( group ) of points that lie on the same line are called collinear points.
Also, if a set of points do not lie on the same line, they are called noncollinear.

Example B

Problem: Given the following points:  C, D, F, & E How many different lines can we make? What are the names of each line? Which points are collinear? Which points are noncollinear?

Solution:

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Example C

Question: What could we say about this:

Solution:

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Plane: 

A plane is a flat surface that:

• Has no thickness
• Extends forever, in any direction on the flat surface from any given point on the plane
• A plane can be identified by either three-noncollinear points or a line in the plane and a point not on the line.
•  A plane is named using either an uppercase letter or any three noncollinear points that lie in the plane
• When drawing a plane, it is usually drawn out as a parallelogram.

Just like points that are on the same line are said to be collinear, points and lines that lie in the same plane are said to be coplanar.
Also, if there is no plane that contains the lines or points, then they are noncoplanar.

Example D

Question: How could we name this plane?

Solution:

Watch the video:

Example E

Problem: Identify the coplanar and noncoplanar lines in the diagram.

Solution:

Watch the video:

Intersections:

An intersection is the Point or set of points ( Line ) in which two figures meet.

• When two lines intersect, their intersection is a single point.
• When two planes intersect, their intersection is a single line. If a line lies in a plane, then their intersection is the line itself. If the line does not lie in the plane, then their intersection is a single point.

Example F

Problem: Identify all of the intersections in this figure.

Solution:

Watch the video:

Practice Problems

Identify the following: a. All of the lines. b. A pair of collinear points. c. All of the planes. d. Three coplanar points. e. Two coplanar lines. f. A pair of. noncoplanar lines	Use this diagram for A - F.
g. What is the intersection of and ?. h. What is the intersection of and plane W? i.  What is the intersection of  and plane W?	Use this diagram for G - I.

Assignment:

Create a figure with the given information from problems 1 -  4. 1.  Create a plane and name it W. 2.  Draw and label coplanar points C, D, and E on the plane. 3.  Draw line CD on the plane. 4.  Place point K on the plane so that it is collinear with point C but not on. 5.  What is another name for plane W ?
6.  Give another name for plane M. 7.  Name three collinear points. 8. Give three points that are noncollinear. 9. Are any of the points noncoplanar? 10. Name two coplanar lines.	Use the following diagram to complete questions 6–10
Create a figure with the given information from problems 11 - 15 11. Create 2-planes A and B, that intersect each other. 12. Draw and label the intersection of planes A and B as . 13. Draw and label so that it intersects plane A at every point       and plane B at point G. 14. Draw and label so that it intersects at point G and       the two lines are not on the same plane. 15. Draw and label point T so that it is coplanar with points Y and Z.
16. What is the intersection of plane J and ? 17. What is the point of intersection of line P and ? 18. What is the intersection of planes J and K ? 19. What is the intersection of  and ? 20. What is the intersection of plane K and ? 21. What is the intersection of plane J and ? 22. What is the intersection of plane K and ?	Use the following diagram to complete questions 16 - 22