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Coordinate Geometry
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1. Points
- Cryptography and Coordinates
- Pick's Theorem
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2. Lines
- Linear Functions - The Basics
- Systems of Linear Equations - Practice
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3. Parabolas
- Quadratic Inequalities (Equations and Parabolas)
- Coefficients of a Parabola - Practice
- Parabolas in Vertex Form - Practice
- How to: Parabola through 3 points
- Orthogonal parabolas - Exploration
- Explore! From 3D to 2D: Parabola as Projection of a Circle
- Parabola - An Optical property
- Parabola as Envelope
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4. Circles
- Circle and Coordinate Geometry - Lesson+Practice
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5. Ellipses
- Ellipse as locus of points - Lesson
- Ellipse as Envelope of Segments
- Ellipse - An Optical Property
- Ellipse as Envelope
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6. Conics
- "All-in-One" Loci: Ellipse, Hyperbola and Circle - Lesson
- Conics and Euclidean Invariants
- Ellipse, Hyperbola and Circle as Envelopes
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7. Transformations
- Matrices and Linear Transformations
- Transformations of Triangles and Matrices
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Coordinate Geometry
Collection of Coordinate Geometry activities.
Table of Contents
- Points
- Cryptography and Coordinates
- Pick's Theorem
- Lines
- Linear Functions - The Basics
- Systems of Linear Equations - Practice
- Parabolas
- Quadratic Inequalities (Equations and Parabolas)
- Coefficients of a Parabola - Practice
- Parabolas in Vertex Form - Practice
- How to: Parabola through 3 points
- Orthogonal parabolas - Exploration
- Explore! From 3D to 2D: Parabola as Projection of a Circle
- Parabola - An Optical property
- Parabola as Envelope
- Circles
- Circle and Coordinate Geometry - Lesson+Practice
- Ellipses
- Ellipse as locus of points - Lesson
- Ellipse as Envelope of Segments
- Ellipse - An Optical Property
- Ellipse as Envelope
- Conics
- "All-in-One" Loci: Ellipse, Hyperbola and Circle - Lesson
- Conics and Euclidean Invariants
- Ellipse, Hyperbola and Circle as Envelopes
- Transformations
- Matrices and Linear Transformations
- Transformations of Triangles and Matrices
Cryptography and Coordinates
Your school prohibits using cellphones during school hours, but you and your friends found a way to communicate using encrypted messages, so that just those who have the key can understand the content of the message.
Here is your key!
It's a grid, containing all the letters of the alphabet. Each letter has its own coordinates, and only who has the same exact grid can decode the message.
The Signature
Every message must be signed, in order to understand who is the sender.
My signature is .
What is yours?
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Separating Words
An encrypted message must be understandable though!
We will separate words by enclosing each one of them in braces.
Write an encrypted message to a classmate.
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Securing Data
Securing data is the core principle on which cryptography methods are based.
If we used as key just the x-coordinates of the points that you see in the grid, would this method be as safe as the original one?
And what if we used only y-coordinates?
If you had to choose between x-coordinates or y-coordinates only to encrypt your message, which of the two methods would be the most efficient and safe?
Explain your choice in detail.
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Create Your Own Decryption Grid
Drag the points in the app below to create your own decryption grid.
Points must respect the following conditions:
- each point must have unique coordinates
- the x-coordinate of the vowels is -1
- the x-coordinate of D is -6
- the x-coordinate of H and K is -5
- points W, X, Y and Z belong to the y-axis
- points L, M, N, R, S and T belong to the x-axis
- the y-coordinate of B, F, J, P and V is 3
- points C, G, and Q have their x-coordinate equal to their y-coordinate
Linear Functions - The Basics
Graph of a linear function and slope
The graph of a function of the form is a line.
This is why all the functions of this type are named linear functions.
If we know the coordinates of two points of the function, and , we can calculate the slope of the line: . This is a constant value: however you choose two points on the line, the value of m is always the same.
Try it yourself...
In the app below, move points and , then enter in the input box the value of the slope of the line that you have defined.
Select Check answer to get a feedback for your answer and view the solution of this exercise.
Deselect Check answer to create a new line and calculate its slope.
When things go wrong algebraically...
If you have the equation of a linear function and the coordinates of two of its points, and , you can calculate:
- the value of the y-intercept
- the value of the slope, using the formula .
Move points A and B in the app above, and align them vertically.
You will discover which is the algebraic issue that is generated by such a configuration.
... and geometrically
Move points A and B in the app above, and align them vertically.
Observe the graph of the line.
Is this the graph of a linear function?
Explain your conjectures.
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This is not the graph of a function, because a function of a variable x is a relation that assigns to each input value x exactly one output value y.
Your example in the app shows that there are at least two values of y assigned to the same value of x (actually, there are infinitely many of them!), and this contradicts the definition of function.
Parabolas
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1. Quadratic Inequalities (Equations and Parabolas)
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2. Coefficients of a Parabola - Practice
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3. Parabolas in Vertex Form - Practice
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4. How to: Parabola through 3 points
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5. Orthogonal parabolas - Exploration
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6. Explore! From 3D to 2D: Parabola as Projection of a Circle
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7. Parabola - An Optical property
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8. Parabola as Envelope
Quadratic Inequalities (Equations and Parabolas)
Solving a Quadratic Inequality Graphically
The graph of a quadratic function in a Cartesian coordinate system is a parabola.
To solve a quadratic inequality or graphically, draw the corresponding parabola, then:
- The x-coordinates of the points (if they exist) where the parabola intersects the x-axis are the solution of the equation (zeros of the equation)
- The x-coordinates of the points (if they exist) where the parabola is above the x-axis are solution of the inequality
- The x-coordinates of the points (if they exist) where the parabola is below the x-axis are solution of the inequality
Ready, Set, Practice!
Find the solutions of a quadratic equation or inequality by exploring the graph of the corresponding parabola.
Use the input box to enter different quadratic expressions and the drop down list to select the equation or inequality form to solve.
Use the mouse wheel or the predefined gestures for mobile devices to zoom in/out and view details in the Graphics View.
Today You Are the Teacher!
Today's assignment is the inequality .
Alice solves it like this: .
Bob solves first, and gets . Then he graphs the parabola corresponding to the given equation and finds the solution 
.
Chuck uses ChatGPT 2 and finds that the solution is .
Alice says that her method is faster than Bob's, because it doesn't require sketching a graph.
Grade your students' solutions, and explain the reasons for your grading.
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Chuck uses ChatGPT 2 and finds that the solution is .
Alice says that her method is faster than Bob's, because it doesn't require sketching a graph.
Grade your students' solutions, and explain the reasons for your grading.Font sizeFont size
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Hamletic Doubt...
Below you can see the solution of one of the following inequalities.
Choose the correct one.
Circle and Coordinate Geometry - Lesson+Practice
Discover how to write the equation of a circle given its center and radius, and how to find center and radius given the general equation of a circle.
Explore the applet, then answer the questions displayed below.
Select the option "Circle - given center and radius" in the app above, then drag the center C at (0,0).
Use the slider to display different circles with center at (0,0).
What is the equation of a circle with center at (0,0) and radius r?
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A circle with center C=(0,0) passes through P=(2,3). What is the equation of the circle?
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(The radius is )
Given A=(-8,-3) and B=(2, 5), write the equation of the circle with diameter AB.
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or
One way to solve it is by considering that if AB is the diameter, the midpoint of the segment AB is the center C of the circle.
CA (or CB, or 1/2 AB) is the radius. Plug the values of center and radius in the equation of the circle given center and radius, and expand the equation.
Select the option "Circle - general equation" in the applet above and use the sliders to explore the equation and the corresponding graph.
When ...
When ...
When ...
When ...
Does the equation represent a circle?
If so, determine its center and radius.
If not, explain why this is not the equation of a circle.
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The graph of this equation is just a point, and precisely point C=(-5,-6).
Consider the general equation of a circle, . In our case , and .
The equation represents a circle is and only if it has a center and a radius.
and
Being the radius 0, the circle is degenerated into its center C.
Ellipse as locus of points - Lesson
Explore the construction of an ellipse, as locus of points having the same property.
Use data shown in the activity to determine the equation of the ellipse.
"All-in-One" Loci: Ellipse, Hyperbola and Circle - Lesson
Move the foci F1and F2 that define the position of the focal axis of the conic, set the length of the major axis of the conic using the mAxis slider, then move point P along the circle to build the locus.
To create the construction:
- Plot 2 points, F1 and F2 that will be the foci of the conic.
- Draw the circle with center F1 and radius r = mAxis
- Create a point P on the circle, then draw the perpendicular bisector of segment PF2
- Line PF1 intersects the perpendicular bisector at a point L, which is a point of the conic
Explore the locus
After creating the locus, draw triangle PLF2 and answer the following questions:
What type of triangle is PLF2?
Explain.
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Write the canonical definition (as locus) of the conic that you see in the construction.
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Use the properties of triangle PLF2 to show that the graph in the construction matches the canonical definition.
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Matrices and Linear Transformations
Explore linear transformations applied to different objects: points, lines, circles, functions and to the unit square with a vertex in the origin.
The starting objects are displayed in green, and can be dragged; the transformed objects are displayed in red.
Some predefined transformations are already available: reflections about axes, shears, homotheties (dilations) and rotations.
You can also select a custom transformation, and define the transformation matrix by selecting the desired values for a, b, c, and d using the corresponding sliders.
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