What if you were given a picture of a figure or an object, like a map with cities and roads marked on it? How could you explain that picture geometrically? After completing this Concept, you'll be able to describe such a map using geometric terms.
CK-12 Foundation: Chapter1BasicGeometricDefinitionsA
James Sousa: Definitions of and Postulates Involving Points, Lines, and Planes
A point is 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.
A line is a set of 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 PQ←→, QP←→ or just g. You would say “line PQ,” “line QP,” or “line g,” respectively. Notice that the line over the PQ←→ and QP←→ has arrows over both the P and Q. The order of P and Q does not matter.
A plane is 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 M. Again, the order of the letters does not matter.
We can use point, line, and plane to define new terms. Space is the set of all points extending 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.
Points that lie on the same line are collinear. P,Q,R,S, and T are collinear because they are all on line w. If a point U were located above or below line w, it would be non-collinear.
Points and/or lines within the same plane are coplanar. Lines h and i and points A,B,C,D,G, and K are coplanar in Plane J. Line KF←→ and point E are non-coplanar with Plane J.
An endpoint is a point at the end of a line segment. Line segments are labeled by their endpoints, AB¯¯¯¯¯ or BA¯¯¯¯¯. Notice that the bar over the endpoints has NO arrows. Order does not matter.
A ray is a part of a line with one endpoint that extends forever in the direction opposite that endpoint. 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, CD−→− or DC←−−.
An intersection is a point or set of points where lines, planes, segments, or rays cross each other.
With these new definitions, we can make statements and generalizations about these geometric figures. This section introduces a few basic postulates. Throughout this course we will be introducing Postulates and Theorems so it is important that you understand what they are and how they differ.
Postulates are basic rules of geometry. We can assume that all postulates are true, much like a definition. Theorems are statements that can be proven true using postulates, definitions, and other theorems that have already been 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 course.
Postulate #1: Given any two distinct points, there is exactly one (straight) line containing those two points.
Postulate #2: Given any three non-collinear points, there is exactly one plane containing those three points.
Postulate #3: If a line and a plane share two points, then the entire line lies within the plane.
Postulate #4: If two distinct lines intersect, the intersection will be one point.
Postulate #5: If two distinct planes intersect, the intersection will be a line.
When making geometric drawings, be sure to be clear and label all points and lines.
What best describes San Diego, California on a globe?
Answer: A city is usually labeled with a dot, or point, on a globe.
Use the picture below to answer these questions.
a) List another way to label Plane J.
b) List another way to label line h.
c) Are K and F collinear?
d) Are E,B and F coplanar?
a) Plane BDG. Any combination of three coplanar points that are not collinear would be correct.
b) AB←→. Any combination of two of the letters A,B, or C would also work.
Describe the picture below using all the geometric terms you have learned.
AB←→ and D are coplanar in Plane P, while BC←→ and AC←→ intersect at point C which is non-coplanar.
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter1BasicGeometricDefinitionsB
A point is an exact location in space. A line is infinitely many points that extend forever in both directions. A plane is infinitely many intersecting lines that extend forever in all directions. Space is the set of all points extending in three dimensions. Points that lie on the same line are collinear. Points and/or lines within the same plane are coplanar. An endpoint is a point at the end of part of a line. A line segment is a part of a line with two endpoints. A ray is a part of a line with one endpoint that extends forever in the direction opposite that point. An intersection is a point or set of points where lines, planes, segments, or rays cross. A postulate is a basic rule of geometry is assumed to be true. A theorem is a statement that can be proven true using postulates, definitions, and other theorems that have already been proven.
1. What best describes the surface of a movie screen?
2. Answer the following questions about the picture.
a) Is line l coplanar with Plane V, Plane W, both, or neither?
b) Are R and Q collinear?
c) What point belongs to neither Plane V nor Plane W?
d) List three points in Plane W.
3. Draw and label the intersection of line AB←→ and ray CD−→− at point C.
4. How do the figures below intersect?
1. The surface of a movie screen is most like a plane.
2. a) Neither
d) Any combination of P,O,T, and Q would work.
3. It does not matter the placement of A or B along the line nor the direction that CD−→− points.
4. 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.
- Name this line in two ways.
- Name the geometric figure below in two different ways.
- Draw three ways three different planes can (or cannot) intersect.
- What type of geometric object is made by the intersection of a sphere (a ball) and a plane? Draw your answer.
Use geometric notation to explain each picture in as much detail as possible.
For 6-15, determine if the following statements are ALWAYS true, SOMETIMES true, or NEVER true.
- Any two distinct points are collinear.
- Any three points determine a plane.
- A line is composed of two rays with a common endpoint.
- A line segment has infinitely many points between two endpoints.
- A point takes up space.
- A line is one-dimensional.
- Any four distinct points are coplanar.
AB−→− could be read “ray AB” or “ray BA.”
AB←→ could be read “line AB” or “line BA.”
- Theorems are proven true with postulates.
In Algebra you plotted points on the coordinate plane and graphed lines. For 16-20, use graph paper and follow the steps to make the diagram on the same graph.
- Plot the point (2, -3) and label it A.
- Plot the point (-4, 3) and label it B.
- Draw the segment AB¯¯¯¯¯.
- Locate point C, the intersection of this line with the x−axis.
- Draw the ray CD−→− with point D(1,4).