Chapter 1 Understanding Points, Lines, and Planes.
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Transcript of Chapter 1 Understanding Points, Lines, and Planes.
Chapter 1
Understanding Points, Lines, and Planes
Geometry
The branch of mathematics that studies the shapes of objects and their sizes, properties and relationships, and spatial reasoning.
Geo - “earth” metry - “measure”– Geometry literally means earth measure
We will study the logical structures of geometry and its relationships to algebra, probability and triogonometry.
Terms MUST KNOW!
*Point – names a location
Has no size.
Use capital letters EX: P, Point P, Pt P, Not: •P (u do not need the • to name a pt)
• P
• Q
*Line – straight path that never ends **
Symbol
to name: AB
BA or line l A B l
*Plane – flat surface that extends forever
To name use the Script capital letter or 3 points not on the same line.
EX: P or ABCP
AB
C
Terms Continued….
Segment – part of a line consisting of 2 pts and all pts in between **
Symbol
CD or DCnot CD (must have segment above letters)
Endpoint – pt at one end of a segment or the starting pt of a ray C or D
look at the above pictures points C and D are endpts
Ray – part of a line that starts at an endpt & extends forever in 1 direction **
Symbol
AB or CDnot BA (the endpt must be the first pt listed)
C D
A B
CD
Terms Continued…
Opposite Ray – 2 rays with a common endpt that form a line
BA & BC
Collinear Pts on the same line
3 non-collinear pts determine a plane
Coplanar Pts on the same plane
If 2 pts are in a plane, then the line containing them is in the plane
A B C
* The most basic figures, points, lines, and planes are “undefined terms” in geometry. This means they cannot be defined using other figures.
** 2 letters without a symbol refers to distance.
you must use the correct symbol to represent either a line, segment, or ray!!
ONLY HONORS NEEDS TO WRITE THIS SLIDE!
The Geometry we are familiar with and the kind we will study is called Euclidean geometry, named after the Greek mathematician, Euclid.
Other geometries, called non-Euclidean– Spherical Geometry – geometry of the sphere. More
suited to our earth. Planes are the surfaces of a sphere and all lines are circles on the sphere
– Hyperbolic Geometry – geometry on a circular plane– Coordinate Geometry – (analytic geometry) uses the
coordinate system (x,y) to study properties of lines, segments, etc.
Examples
1. Name the two planes.2. What is the intersection of the two planes?3. Which plane contains points G, E and D? 4. Which plane contains point H?5. What points are not in both planes?6. Name a point not in plane N.7. Point F is the intersection of _ ____ and ______.
M & N
AB
MN
G, E, D, C, H, WG
AB CH
Examples
True or False8. B, E, and D are coplanar9. G, A, and F are collinear10. H and E determine a line11. C & H determine plane N12. ray AF and ray FB are opposite rays13. Line AB and Plane M have exactly 3 points of intersection.
True
False
True
False
False
False
Optical Illusions
Your eyes see images and send them to your brain to interpret, but sometimes the brain can be tricked and your spatial reasoning is tested.
Optical Illusion Examples
Are the petals moving?
you will probably find it hard to believe that squares A and B are actually the same shade of gray
Honors Only!!
Slope formula y2-y1 = m
x2-x1
y = mx + b y – y1 = m(x – x1) Parallel lines have SAME slope DIFF y- intercepts Perpendicular lines’ slopes are opposite reciprocals
example : 3 and -1/3 Oblique lines have DIFFERENT slopes, not opposite
reciprocals