Point, line, plane. Basics of Geometry in Mathematics 7.ppt
- 1. Basics of Geometry Basics of Geometry POINTS! LINES! PLANES! OH MY!
- 2. Basics of Geometry • What are the undefined terms in geometry? • What concepts present the foundations of geometry? • Can you sketch the intersection of lines and planes? These questions (and much more!!) will be answered by the end of this presentation. Are you ready?
- 3. Basics of Geometry The terms points, lines, and planes are the foundations of geometry, but… point, line, and plane are all what we call undefined terms. How can that be? Well, any definition we could give them would depend on the definition of some other mathematical idea that these three terms help define. In other words, the definition would be circular! Undefined Terms?
- 4. Basics of Geometry • Has no dimension • Usually represented by a small dot Point A The above is called point A. Note the point is represented with a capital letter.
- 5. Basics of Geometry • Extend in one dimension. • Represented with straight line with two arrowheads to indicate that the line extends without end in two directions. Line l B A This is Line l, (using the lower case script letter) or symbolically we call it AB NOTICE: The arrowheads are in both directions on the symbol AB
- 6. Basics of Geometry Plane • Extend in two dimensions. • Represented by a slanted 4 sided figure, but you must envision it extends without end, even though the representation has edges. A B C M This is Plane M or plane ABC (be sure to only use three of the points when naming a plane)
- 7. Basics of Geometry Undefined Concepts • Collinear points are points that lie on the same line. l C A B Points A, B and C are collinear.
- 8. Basics of Geometry Undefined Concepts • Coplanar points are points that lie on the same plane. A B C Points A, B and C are coplanar.
- 9. Basics of Geometry Line Segment Let’s look at the idea of a point in between two other points on a line. Here is line AB, or recall symbolically AB A B The line segment does not extend without end. It has endpoints, in this case A and B. The segment contains all the points on the line between A and B A B Notice the difference in the symbolic notation! This is segment AB
- 10. Basics of Geometry Symbolized by Ray AB The initial point is always the first letter in naming a ray. Notice the difference in symbols from both a line and segment. A B A is called the initial point Ray AB extends in one direction without end. Let’s look at a ray:
- 11. Basics of Geometry Not all symbols are created equal! Symbol alert! BA AB is the same as AB BA is the same as BUT… A B A B
- 12. Basics of Geometry AB is not the same as BA Symbol alert!! A B AB A B BA Initial point 1st The ray is different! Notice that the initial point is listed first in the symbol. Also note that the symbolic ray always has the arrowhead on the right regardless of the direction of the ray.
- 13. Basics of Geometry If C is between A and B, Opposite Rays CA A B C then and are opposite rays. CB C is the common initial point for the rays!
- 14. Basics of Geometry Rays are important because they help us define something very important in geometry…Angles! An angle consists of two different rays that have the same initial point. The rays are sides of the angles. The initial point is called the vertex. Angles vertex sides A B C Notation: We denote an angle with three points and symbol. The middle point is always the vertex. We can also name the angle with just the vertex point. This angle can be denoted as: A or CAB, , BAC
- 15. Basics of Geometry Straight angle m A = 180° Obtuse angle 90°< m A < 180° Acute angle 0°< m A < 90° Classifying Angles Angles are classified as acute, right, obtuse, and straight, according to their measures. Angles have measures greater than 0° and less or equal to 180°. A A Right angle m A = 90° A A
- 16. Basics of Geometry • Two or more geometric figures intersect if they have one or more points in common. • The intersection of the figures is the set of points the figure has in common Intersections of lines and planes Think! ! How do 2 line intersect? How do 2 planes intersect? What about a line and a plane?
- 17. Basics of Geometry Modeling Intersections To think about the questions on the last slide lets look at the following… B A E Two lines intersect at a point, like here at point A. F Line BF is the intersection of the planes G and H. G H Point E is the intersection of plane H and line EC D C
- 18. Basics of Geometry Something to think about… You have just finished the first section in Geometry! This is a very important section because it lays the foundation for the rest of the year! Much of the vocabulary you will encounter in this course will have its foundation in the ideas presented in this lesson. Can you name the three undefined terms in geometry? Do you know the difference between and obtuse and straight angle? Can you sketch the intersection of a plane and a line? How about two planes? Can you visualize the intersection of two planes? How about three? The classfun and homefun provided will help you in developing a better understanding of the concepts!