Christmas Wreath

Drag the sliders to create different Christmas wreaths.

Irina Boyadzhiev, Dec.8, 2015, Created with GeoGebra
Download applet

On the Geometric Definition of Ellipse

The applet demonstrates the following:
An ellipse is the set of all points in the plane, the sum of whose distances to two fixed points (foci) remains constant. 

• Select the length of a piece of string by dragging the endpoints of the blue segment. 
• Drag the orange point to select the position of the focus F1 along the x-Axis or the y-Axis. The other focus F2 is symmetrical to F1 with respect to the origin. 

A string with the selected length is attached to both foci and is kept tight by the tip of the pencil.

• Drag the tip of the pencil or press the “Draw” button to trace all points on the plane that satisfy the above definition.
•  Hide the pencil by pressing the "Pencil ON/OFF" button; show the ellipse by pressing “Show Ellipse” button, and explore the curve by changing the positions of the foci and the length of the constructing string. 
•  Bring the two foci to the origin to see the circle as a special case of the ellipse. 
• Click on “Labels” to see some terminology.

Irina Boyadzhiev, April 19, 2015, Created with GeoGebra

Download applet
Irina Boyadzhiev's  GeoGebra Applets

Radians, Number Line and the Unit Circle

This applet demonstrates the correspondence between the points on the number line and the points on the unit circle.

• Enter a real number in the “step” input box. Try the applet with whole numbers, fractions, multiples or fractions of pi. (Example: pi/4)
• Select “Wrap Positive Numbers” or “Wrap Negative Numbers”.

Notice, you have to deselect one checkbox in order to be able to select the other one.

Irina Boyadzhiev, January 9, 2015, Created with GeoGebra

Download applet
Irina Boyadzhiev's  GeoGebra Applets

Applet on Counting the Number of Squares in a Chessboard

This applet models how to count the number of squares in the chessboard.

Midsegmens and Congruent Triangles

A midsegment of a triangle is a segment that connects the midpoints of any two sides of the triangle. This applet demonstrates that the three midsegments of any triangle divide the original triangle in four congruent triangles. The demonstration is based on the following properties:

1. The midsegment is parallel to the third side of the triangle. 
2. Any parallelogram has a rotational symmetry with center the intersection point of the diagonals and angle of rotation 180∘

Irina Boyadzhiev, 9 February 2014, Created with GeoGebra

Download applet
Irina Boyadzhiev's  GeoGebra Applets

The Area of the Median Triangle - Dynamic Proof

Let ABC be a given triangle. The triangle with sides equal to the medians of ABC is called the Median Triangle.
We will show that the area of the median triangle is 3/4 of the area of the original triangle ABC.

•  Drag the two sliders to the end to construct the median triangle by translating the medians.

To rearrange the area of the median triangle:

• Click the Rearrange checkbox. This will show a green point at the vertex of the median triangle. 
•  Drag the green point to C.
•  Drag the green point at B to M_c. 
• Drag the green point at M_a to M_c. 

Irina Boyadzhiev, 29 January 2014, Created with GeoGebra

Download applet
Irina Boyadzhiev's  GeoGebra Applets

Pythagorean Theorem

Irina Boyadzhiev, 13 December 2013, Created with GeoGebra

Download applet
Irina Boyadzhiev's  GeoGebra Applets

Vertical Line Test

This applet can be used to determine whether one relation is a function or not by using the Vertical Line Test. The Vertical Line Test says that if some vertilcal line meets the graph in more than one point then the relation is not a function.

•  In the input box enter a polynomial equation of x and y in the form "polynomial = number".
  For example, to graph y = x², enter y - x² = 0 
• Drag the slider to use the vertical line test.

Irina Boyadzhiev, 25 September 2013, Created with GeoGebra

Download applet
Irina Boyadzhiev's  GeoGebra Applets

Inverse Function

This applet can be used to study inverse functions.

• Enter a function of x in the input box f(x).
• Set the domain of the function by dragging the endpoints of the blue line at the bottom of the window.
• Click on the “Horizontal Line Test” and drag the vertical slider. Is f(x) a one-to-one function?
• Click on the “Reflect f(x)” button.
• Drag the horizontal slider to run the Vertical Line Test. Is the reflection a graph of a function?
• Drag the endpoints of the blue segment under the graphs to restrict the domain of f(x) in such a way that the reflected graph is a graph of a function.
• Hide all points created by the horizontal and vertical line tests by clicking the respective checkboxes.
• Click the “Show Corresponding Points”. Drag the orange point along the graph of f(x) and notice its reflection on the graph of the inverse function.

Irina Boyadzhiev, 25 September 2013, Created with GeoGebra

Download applet 

Irina Boyadzhiev's  GeoGebra Applets

Horizontal Line Test

This applet can be used to determine whether a function is one-to-one or not, and also to restrict the domain to some interval where the function is one-to-one.

• Enter a function of x in the input box. 
• Drag the vertical slider up or down (or press the Play button) to find the intersecting points of the graph and the horizontal line. 
• To see the x-coordinates of the intersection points, check the “show coordinates”. 

If we want to define an inverse function then we have to find an interval where f(x) is one-to-one.

•  Drag the left and/or the right side of the blue segment under the graph to restrict the domain.

Use the second button on the toolbar to drag the viewing window, zoom in or zoom out if necessary.

Irina Boyadzhiev, 19 September 2013, Created with GeoGebra

Download applet 

Irina Boyadzhiev's  GeoGebra Applets

The Sum of the Cubes of the First N Natural Numbers Dynamic Geometric Proof

This applet shows a dynamic geometric proof of the formula for the sum of the cubes of the first n natural numbers.

• Drag the slider “NumberCubes” to create up to seven cubes.

Think of each cube as a collection of n nxn square layers of unit cubes. To find a formula for the sum of the cubes of the first natural numbers we will rearrange the square layers.

• Drag the slider “Rearrange” or press the “animate” button.

Irina Boyadzhiev, 29 April 2013, Created with GeoGebra

Download applet 

Irina Boyadzhiev's  GeoGebra Applets

Octagon to Square Dissection

The applet below shows the octagon to square dissection.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com Reference:
"Bennett's Octagon-to-Square Dissection" from the Wolfram Demonstrations Project http://demonstrations.wolfram.com/BennettsOctagonToSquareDissection/ Contributed by: Izidor Hafner Based on work by: Greg N. Frederickson

Square Root of a Sum? - Geometric Interpretation

We know that for non-negative numbers  Can we apply the same for the sum? 
In the example below a and b are the legs of a right triangle.  According to the Pythagorean Theorem the hypotenuse is

• Drag to the left the blue point A to rotate the leg AC around point C until the two legs form one segment.
  
• Drag to the left the orange point A to rotate the hypotenuse around point B until it comes to the base of the triangle.
• Compare the lengths  

Irina Boyadzhiev, 6 April 2013, Created with GeoGebra

Download applet 

Irina Boyadzhiev's  GeoGebra Applets

The Sum of the First n Natural Numbers - Dynamic Geometric Proof

This applet gives a dynamic proof of the formula for the sum of the first n natural numbers.


Irina Boyadzhiev, 6 March 2013, Created with GeoGebra

Download the file

• See the general case - Arithmetic Series.

Irina's GeoGebra Applets

The Sum of Squares - dynamic proof

This applet gives a  dynamic proof of the formula for the sum of the squares of the first n natural numbers.  We start with three times the sum of the squares and rearrange the parts of one of the sums.

• Click on "Color the third column".
• Drag the slider to rearrange the parts of the third column.

The area of the formed rectangle equals three times the sum of the squares.

• Click on  Show the formulas  to see the result.

Irina Boyadzhiev, 13 February 2013, Created with GeoGebra

Download the file

Irina's GeoGebra Applets

Domain and Range of a Function by Flattening to the Axis

This applet can be used to introduce the concept of Domain and Range of a function by flattening the graph of the function over the coordinate axes.

• Click on a button to select one of the three functions.
• Drag the "domain" slider. The graph of the function is flattened to the x-axis. The trace that is left on the x-axis is the domain of the function.
• Drag the "range" slider. The graph of the function is flattened to the y-axis.  The trace left on the y-axis is the range of the function. 

Download File

Irina Boyadzhiev - GeoGebra Applets

Domain and Range of a Function

This applet can be used to demonstrate the concept of Domain and Range of a function.

• Click on a button to select one of the three functions.
• Drag the "domain" slider. A vertical ray will scan the viewing window and will project the intersection point of the ray and  the function to the x-axis. The domain appears as red interval(s) on the x-axis.
• Drag the "range" slider. A horizontal ray will scan the viewing window and will project its intersection  points with the function to the y-Axis. The range appears as green interval(s) on the y-axis.

Irina Boyadzhiev, 12 January 2013, Created with GeoGebra
Download Applet
Irina Boyadzhiev - GeoGebra Applets

Square Octagon Tessellation

One of the objectives of GeoGebra Applet Central this year is to update the output files of my GeoGebra Tutorial Series. The previous outputs were still in version 3.2. The applet below is the output of GeoGebra Tutorial 10 - Vectors and Tessellation. I just changed the color of the vectors to make it more visible. In this tutorial, the Vector between Two Points tool is used to translate objects and create a tessellation.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com The Step by Step instructions on how to create the applet above can be found here.

Year 2012 in Review - The Most Popular Posts

It's the end of the year and it's time to look back your favorite posts. Below are the most popular posts for 2012 in terms of traffic (Source: Google Analytics).

1. GeoGebraTube: 12000+ applets and counting
2. Pythagorean Theorem Proof
3. Animation Using Bezier Curves
4. Math and Multimedia Carnival 20
5. The Arithmetic-Mean Geometric-Mean Inequality
6. Deriving the Equation of the Parablola
7. Fibonacci Number and the Fibonacci Spiral
8. Tangram Puzzle: Constructing a Square
9. Systems of Linear Equations
10. The Euler Line
11. Bezier Curves - How Recursive Algorithm Works
12. Squaring a Triangle

Have a prosperous new year everyone!

Square Rectangle Tessellation

The applet below is another tessellation. It is composed of rectangles and equilateral triangles. It is one of the demiregular tessellations.
This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com