Class : 9th
Maths activity :
1) Explain all five Postulates of Euclid's.
Maths project :
1) Draw the graph of the line x+4=0, y+3=0.
Explain where the both lines intersect.
Explain which line is parallel to the x-axis and which is parallel to the y-axis
..
plz answer its given by my sir .
don't give spam answer
Answers
Answer:
The New Course: Geometric Structures for the Visual
[Work in Progress DRAFT VERSION Based on noted from 09]
Blue sections indicate tentative plans for those dates.
Tuesday
Tues/Thursday Crossover
Thursday
Introduction
Continue discussion of what is "geometry"?
Start on Euclid- Definitions, Postulates, and Prop 1.
Euclid- Definitions, Postulates, and Prop 1.
Pythagorean plus... Dissections ?
Dissections- equidecomposeable polygons
Assign viewing of looking at Euclid
Begin Constructions and the real number line. M&I's Euclidean Geometry
More on Equidecomposeable polygons
Construction of rational numbers. Constructions and The real number line.- Continuity Inversion and Orthogonal Circles
.
More on Inversion. and Continuity. Similar Triangles
Odds and ends, The real number line.- Continuity. Coordinate based proofs.
.Isometries:Classification of Isometries & More :)
Isometries. Coordinates and Classification
Proof of classification result for plane isometries
More Isometries
Recognizing Isometries
Symmetry.
Similarity & Proportion (Euclid V) More on Proportion and Measurement More on Similarity and transformations.
Similarity.
Inversion and Beginning to See The Infinite.
The Affine Line and Homogeneous Coordinates.
Homogeneous Coordinates
More on seeing the infinite.
The Affine Line and Homogeneous Coordinates.
No class Spring Break!
No class Spring Break!
No class Spring Break!
3-23 More on The Affine Line and Homogeous coordinates and the Affine Plane
Homogeneous Coordinates and the Affine Plane
Introduction to projective geometry with homogeneous coordinates.Finite geometries!
Axioms
Connecting Axioms to Models.
Z2 and Finite Projective Geometry.
Models for Affine and Projective Geometries.
Introduction to Desargues' Theorem- a result of projective geometry
Axioms for Synthetic Projective Geometry (see M&I)
homogeneous functions and coordinates: the circle and parabola in the affine and projective plane from equations. A beginning to conics.
Examples of proofs in synthetic projective geometry.
. Desargues' Theorem-
Quiz 2 Proof of Desargues' Theorem in the Plane
Introduction to duality.
A look at duality and some applications.
Sections and Perspectively related Figures
Some key configurations. space duality, perspective reconsidered.
Space Duality and polyhedra
Sections in Space
Perspectivies as transformations.
The complete Quadrilateral.
Inversion Video?
More on Perspectivities and "mapping figures"
Projectivities
Conics. Introduction to Pascal's Theorem
Start Isometries with Homog. Coord.
Matrices
More on Matrix Projective Transformations?
Harmonics: uniqueness and coordinates for Projective Geometry.
More on Harmonics.
Projectivities. Projective relations
Projective Line transformations: Synthetic Projectivities;
Harmonics Theroems
Quiz #3
Projective Conics Video?
Pascal and Brianchon's theorem.
Proof of Brianchon's theorem
Planar transformations and Matrices
An Inversion Excursion?
Introductory Class.
What are different aspects of geometry? How is the study of geometry organized? Analytic(numbers) , Synthetic(axiomatic), Transformations (functions) are three ways to organize information and the study of geometry. Also Projective and Differential geometry were mentions as altenative focuses for studying geometry.
Course Description.
Additional Relevant Notes:
Different types of geometry:
Euclidean: Lengths are important
Similarity: Shape is important
Affine: Parallel lines are important.
Projective: "Shadows" are important
Differential: Curvature is important.
Topological: General shape- especially holes and connectedness- is important.
What is synthetic geometry? A geometry that focuses on connecting statements (theorems, constructions) to a foundation of "axioms" by using proofs.
What is analytic geometry? A geometry that focuses on connecting statements (theorems, constructions) to a foundation of number based algebra.
What is "structural geometry"? A geometry that focuses on connecting statements (theorems, constructions) to a foundation of structures (relations and operations) on sets by using proofs.
Answer:
What Are Parallel Lines?
Parallel lines are two lines that are always the same distance apart and never touch. In order for two lines to be parallel, they must be drawn in the same plane, a perfectly flat surface like a wall or sheet of paper.
Parallel lines are useful in understanding the relationships between paths of objects and sides of various shapes. For example, squares, rectangles, and parallelograms have sides across from each other that are parallel.
In formulas, parallel lines are indicated with a pair of vertical pipes between the line names, like this:
AB || CD
Each line has many parallels. Any line that has the same slope as the original will never intersect with it. Lines that would never cross, even if extended forever, are parallel.
Railroad Tracks and Parallel Lines
Sometimes it is helpful to think of parallel lines as a set of railroad tracks. The two rails of the track are created for the wheels on each side of the train to travel along. Because the wheels of the train are always the same distance apart, they do not get closer, not even when they turn.
Parallel lines with equal distances between
The tracks have to be the same distance apart everywhere. Since they are made by humans, railroad tracks aren't quite parallel, but in order to work properly, they have to be awfully close.
The other difference between railroad tracks and perfectly parallel lines is that tracks are built over hills and valleys. The ground they cover is not perfectly flat. When mathematicians imagine parallel lines, on the other hand, they draw them on a perfectly flat surface.
Graphing Parallel Lines
When two parallel lines are graphed, they must always be at the same angle, which means they'll always have the same slope, or steepness.
Here's a graph of two parallel lines:
Graph showing parallel lines
Even though these two lines don't start in the same place, you can see that they are equally steep. They decrease or slope downward at the same rate. That ensures that they're parallel.
If you look at the equations of those two lines, you may notice something: they are exactly the same except for the numbers on the right hand side, the '6' and '12.' We can use some simple algebra and rewrite the two equations in slope-intercept form (y = mx + b), a form of a line that is the most familiar to people.
The blue line's equation will then be:
y = -3/2x + 6
While the red line will have an equation of:
y = -3/2x + 3
Again, you might notice that the two equations are exactly the same except for one thing: the number to the right of x. That number is called the constant and it tells us how high or low the line sits on the graph. When the line is in slope-intercept form, that number also tells us the y-intercept which is exactly where the line hits the y-axis, the vertical line marking zero on the graph. In this graph, the two lines are exactly the same except that one of them is above the other.
To unlock this lesson you must be a Study.com Member.
Create your account
Additional Activities
Additional Practice with Parallel Lines
In the following practice problems, students will determine whether lines are parallel or not by comparing the slopes of the lines. Students will need to convert the equations of lines to slope-intercept form as well as finding the slopes of lines by using points on a graph.
Practice Problems
1. Are the lines y = 4x + 4 and y = -4x - 4 parallel? If not, what would the slope of the second line need to be to be parallel to the first line?
2. Determine if the lines 3x - 5y = 2 and -15x + 25y = 4 are parallel. If not, what would the slope of the second line need to be to be parallel to the first line?
3. Do the lines in the graph below appear to be parallel?
Use the formula
for calculating the slope between two points to determine if the lines are parallel or not.
Solutions
1. These lines are not parallel since they do not have the same slope. The slope of the first line is 4 and the slope of the second line is -4. In order to be parallel, the slope of the second line would need to be 4.
2. We can find the slopes of the lines by converting to slope-intercept form. We have
and