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Coordinate Geometry

Collection of Coordinate Geometry activities.

Points
1. Cryptography and Coordinates
2. Pick's Theorem
Lines
1. Linear Functions - The Basics
2. Systems of Linear Equations - Practice
Parabolas
1. Quadratic Inequalities (Equations and Parabolas)
2. Coefficients of a Parabola - Practice
3. Parabolas in Vertex Form - Practice
4. How to: Parabola through 3 points
5. Orthogonal parabolas - Exploration
6. Explore! From 3D to 2D: Parabola as Projection of a Circle
7. Parabola - An Optical property
8. Parabola as Envelope
Circles
1. Circle and Coordinate Geometry - Lesson+Practice
Ellipses
1. Ellipse as locus of points - Lesson
2. Ellipse as Envelope of Segments - Lesson
3. Ellipse - An Optical Property
4. Ellipse as Envelope
Conics
1. "All-in-One" Loci: Ellipse, Hyperbola and Circle - Lesson
2. Conics and Euclidean Invariants - Lesson
3. Matrices and Linear Transformations - Lesson+Practice
4. Ellipse, Hyperbola and Circle as Envelopes

Points

1. Cryptography and Coordinates
2. Pick's Theorem

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.[br][br]Here is your key![br]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.[br]My signature is [math]\left(8,5\right),\left(2,-1\right),\left(7,-2\right),\left(0,0\right),\left(8,-6\right),\left(-9,5\right)[/math].[br]What is yours?

Separating Words

An encrypted message must be understandable though! [br]We will separate words by enclosing each one of them in braces.[br][math]\left\{\left(8,5\right),\left(0,0\right),\left(-9,2\right),\left(8,-6\right),\left(8,2\right)\left(8,5\right)\right\}\left\{\left(-6,1\right),\left(0,0\right),\left(0,0\right),\left(8,2\right)\right\}![/math][br]Write an encrypted message to a classmate.

Securing Data

Securing data is the core principle on which cryptography methods are based.[br]If we used as key just the[i] x[/i]-coordinates of the points that you see in the grid, would this method be as safe as the original one?[br]And what if we used only [i]y[/i]-coordinates?[br]If you had to choose between [i]x[/i]-coordinates or y-coordinates only to encrypt your message, which of the two methods would be the most efficient and safe?[br]Explain your choice in detail.

Create Your Own Decryption Grid

Drag the points in the app below to create your own decryption grid.[br][br]Points must respect the following conditions:[br]- each point must have unique coordinates [br]- the [i]x[/i]-coordinate of the [i]vowels [/i]is -1[br]- the [i]x[/i]-coordinate of [i]D[/i] is -6[br]- the [i]x[/i]-coordinate of [i]H[/i] and [i]K[/i] is -5[br]- points [i]W[/i], [i]X[/i], [i]Y[/i] and [i]Z[/i] belong to the [i]y-axis[/i][br]- points [i]L[/i], [i]M[/i], [i]N[/i], [i]R[/i], [i]S[/i] and [i]T[/i] belong to the [i]x-axis[/i][br]- the [i]y[/i]-coordinate of [i]B[/i], [i]F[/i], [i]J[/i], [i]P[/i] and [i]V[/i] is 3[br]- points [i]C[/i], [i]G[/i], and [i]Q[/i] have their [i]x[/i]-coordinate equal to their [i]y[/i]-coordinate[br]

1.2.

2.

Lines

1. Linear Functions - The Basics
2. Systems of Linear Equations - Practice

2.1.

Linear Functions - The Basics

Graph of a linear function and slope

The graph of a function of the form [math]y=mx+b[/math] is a [i]line[/i]. [br]This is why all the functions of this type are named [i]linear functions[/i].[br][br]If we know the coordinates of two points of the function, [math]P_1=\left(x_1,y_1\right)[/math] and [math]P_2=\left(x_2,y_2\right)[/math], we can calculate the [i]slope[/i] [math]m[/math] of the line: [math]m=\frac{y_2-y_1}{x_2-x_1}[/math]. This is a constant value: however you choose two points on the line, the value of [i]m[/i] is always the same.

Try it yourself...

In the app below, move points [math]A[/math] and [math]B[/math], then enter in the input box the value of the slope [math]m[/math] of the line that you have defined.[br]Select [i]Check answer[/i] to get a feedback for your answer and view the solution of this exercise.[br]Deselect [i]Check answer[/i] to create a new line and calculate its slope.

When things go wrong algebraically...

If you have the equation of a linear function [math]f\left(x\right)=mx+b[/math] and the coordinates of two of its points, [math]P_1=\left(x_1,y_1\right)[/math] and [math]P_2=\left(x_2,y_2\right)[/math], you can calculate: [br]- the value [math]b[/math] of the [i]y[/i]-intercept[br]- the value [math]m[/math] of the slope, using the formula [math]m=\frac{y_2-y_1}{x_2-x_1}[/math].[br][br]Move points [i]A[/i] and [i]B[/i] in the app above, and align them vertically.[br]You will discover which is the algebraic issue that is generated by such a configuration.

... and geometrically

Move points [i]A[/i] and [i]B[/i] in the app above, and align them vertically.[br]Observe the graph of the line.[br]Is this the graph of a [i]linear function[/i]?[br]Explain your conjectures.

2.2.

3.

Parabolas

1. Quadratic Inequalities (Equations and Parabolas)
2. Coefficients of a Parabola - Practice
3. Parabolas in Vertex Form - Practice
4. How to: Parabola through 3 points
5. Orthogonal parabolas - Exploration
6. Explore! From 3D to 2D: Parabola as Projection of a Circle
7. Parabola - An Optical property
8. Parabola as Envelope

3.1.

Quadratic Inequalities (Equations and Parabolas)

Solving a Quadratic Inequality Graphically

The graph of a [i][color=#1e84cc][b]quadratic function[/b][/color][/i] [math]y=ax^2+bx+c[/math] in a Cartesian coordinate system is a [i][color=#1e84cc][b]parabola[/b][/color][/i].[br][br]To [i][b][color=#1e84cc]solve[/color][/b][/i] a [i][b][color=#1e84cc]quadratic inequality[/color][/b][/i] [math]ax^2+bx+c>0[/math] or [math]ax^2+bx+c<0[/math] [i][b][color=#1e84cc]graphically[/color][/b][/i], draw the corresponding parabola, then:[br][list][*]The[color=#1e84cc] [i][b]x-coordinates[/b][/i][/color] of the points (if they exist) where the [i][b][color=#1e84cc]parabola intersects[/color][/b][/i] the[i][b] [color=#1e84cc]x-axis[/color][/b][/i] are the [i][b][color=#1e84cc]solution[/color][/b][/i] of the[color=#1e84cc] [i][b]equation[/b][/i][/color] [math]ax^2+bx+c=0[/math] (zeros of the equation)[/*][/list][list][*]The [color=#1e84cc][i][b]x-coordinates[/b][/i][/color] of the points (if they exist) where the [i][b][color=#1e84cc]parabola[/color][/b][/i] is[color=#1e84cc] [/color][i][b][color=#1e84cc]above[/color] [/b][/i]the [i][b][color=#1e84cc]x-axis[/color][/b][/i] are [i][b][color=#1e84cc]solution[/color] [/b][/i]of the [i][b][color=#1e84cc]inequality[/color][/b][/i] [math]ax^2+bx+c>0[/math][/*][/list][list][*]The[color=#1e84cc] [i][b]x-coordinates[/b][/i][/color] of the points (if they exist) where the [i][b][color=#1e84cc]parabola[/color][/b][/i] is [i][b][color=#1e84cc]below[/color][/b][/i] the [i][b][color=#1e84cc]x-axis[/color][/b][/i] are [b][i][color=#1e84cc]solution [/color][/i][/b]of the[color=#1e84cc] [i][b]inequality[/b][/i][/color] [math]ax^2+bx+c<0[/math][br][/*][/list]

Ready, Set, Practice!

Find the solutions of a quadratic equation or inequality by exploring the graph of the corresponding parabola.[br][br]Use the input box to enter different quadratic expressions and the drop down list to select the equation or inequality form to solve. [br][br]Use the mouse wheel or the predefined gestures for mobile devices to zoom in/out and view details in the [i]Graphics View[/i].[br][br]

Today You Are the Teacher!

Today's assignment is the inequality [math]x^2-49<0[/math]. [br][br]Alice solves it like this: [math]x^2<49\rightarrow x<\pm7[/math].[br][br]Bob solves [math]x^2-49=0[/math] first, and gets [math]x=\pm7[/math]. Then he graphs the parabola corresponding to the given equation and finds the solution [img]data:image/png;base64,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[/img].[br][br]Chuck uses ChatGPT 2 and finds that the solution is [math]-7\le x\le7[/math].[br][br]Alice says that her method is faster than Bob's, because it doesn't require sketching a graph.[br][br]Grade your students' solutions, and explain the reasons for your grading.

Hamletic Doubt...

Below you can see the solution of one of the following inequalities. [br]Choose the correct one.[br][img]data:image/png;base64,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[/img]

[math]-4+x^2>0[/math] [math]-x^2-4x\le0[/math] [math]-x^2+4x>0[/math] [math]x^2-4>0[/math]

3.2.

3.3.

3.4.

3.5.

3.6.

3.7.

3.8.

4.

Circles

1. Circle and Coordinate Geometry - Lesson+Practice

4.1.

5.

Ellipses

1. Ellipse as locus of points - Lesson
2. Ellipse as Envelope of Segments - Lesson
3. Ellipse - An Optical Property
4. Ellipse as Envelope

5.1.

Ellipse as locus of points - Lesson

Explore the construction of an ellipse, as locus of points having the same property.[br][br]Use data shown in the activity to determine the equation of the ellipse.

5.2.

5.3.

5.4.

6.

Conics

1. "All-in-One" Loci: Ellipse, Hyperbola and Circle - Lesson
2. Conics and Euclidean Invariants - Lesson
3. Matrices and Linear Transformations - Lesson+Practice
4. Ellipse, Hyperbola and Circle as Envelopes

6.1.

"All-in-One" Loci: Ellipse, Hyperbola and Circle - Lesson

Move the foci [color=#9900ff][i]F[/i][sub]1[/sub][/color]and[color=#9900ff] [i]F[/i][sub]2[/sub][/color] that define the position of the focal axis of the conic, set the length of the [i]major axis[/i] of the conic using the [color=#1551b5][i]mAxis[/i][/color] slider, then move point [i][color=#6aa84f]P[/color][/i] along the circle to build the locus.[br][br]To create the construction:[br][list][*]Plot 2 points,[color=#9900ff][i] F[sub]1[/sub][/i][/color][color=#ff0000] [/color]and [i][color=#9900ff]F[/color][/i][sub][i][color=#9900ff]2[/color][/i] [/sub]that will be the foci of the conic.[br][/*][*]Draw the circle with center [color=#9900ff][i]F[/i][sub]1[/sub][/color] and [color=#1e84cc]radius [i]r[/i] = [i]mAxis[/i][/color][/*][*]Create a point [color=#6aa84f][i]P[/i][/color] on the circle, then draw the perpendicular bisector of segment [i]P[/i][i]F[/i][sub]2[/sub][/*][*]Line [i]P[/i][color=#ff0000][color=#000000][i]F[/i][/color][sub][color=#000000]1[/color] [/sub][/color]intersects the perpendicular bisector at a point [color=#6aa84f][i]L[/i][/color], which is a point of the conic[/*][/list][br]Point [color=#6aa84f][i]P[/i][/color], moving on the circle, creates the locus of points of [color=#38761D][i]L[/i][/color], that is:[br][br]- an [i]ellipse [/i]if [color=#9900ff][i]F[/i][sub]2[/sub][/color] is [i]inside [/i]the circle [i] distance between foci[/i] < [i]axis length[/i] → [i]e[/i] < 1[br][br]- an [i]hyperbola [/i]if [color=#9900ff][i]F[/i][sub]2[/sub][/color][color=#ff0000][sub] [/sub][/color]is [i]outside [/i]the circle [i] distance between foci[/i] > [i]axis length[/i] → [i]e[/i] > 1[br][br]- a [i]circle[/i] if [color=#9900ff][i]F[/i][sub]1[/sub] ≡[/color][color=#9900ff][i]F[/i][sub]2[/sub] [/color] [i] distance between foci[/i] = 0 → [i]e[/i] = 0[br][br]([i]e[/i] = [i]eccentricity[/i])

Explore the locus

After creating the locus, draw triangle [i]PLF[sub]2[/sub][/i] and answer the following questions:

What type of triangle is [i]PLF[sub]2[/sub][/i]?[br]Explain.

Write the canonical definition (as locus) of the conic that you see in the construction.

Use the properties of triangle [i]PLF[/i][sub]2[/sub] to show that the graph in the construction matches the canonical definition.

0