Monday, July 30, 2012

The GeoGebra Pantograph

A pantograph is a mechanical linkage connected in a special manner based on parallelograms so that the movement of one pen, in tracing an image, produces identical movements in a second pen. If a line drawing is traced by the first point, an identical, enlarged or miniaturized copy will be drawn by a pen fixed to the other. The pantograph was invented by Cristoph Scheiner.

This is a Java Applet created using GeoGebra from - it looks like you don't have Java installed, please go to
By Guillermo Bautista Jr. , Created with GeoGebra

The applet above demonstrates how the pantograph works.  Move the blue point (hinge) and draw any shape. 

The applet above demonstrates how the pantograph works.  Move the blue point (hinge) and draw any shape. 

1.) What do you observe?
2.) What is the relationship between the blue figure and red figure?
3.) Justify your answer in 2.
4.) Prove that the relationship in 2 always holds.

Sunday, July 29, 2012

Approximating Pi

Move slider n to the right and observe what happens to the inscribed polygon.
This is a Java Applet created using GeoGebra from - it looks like you don't have Java installed, please go to
By Guillermo Bautista Jr. , Created with GeoGebra

The applet above shows the relationship between the area of a circle and the inscribed polygon.  Notice that as the the number of sides of the inscribed polygon (n) increases, its area gets closer and closer to the area of the circle which is equal to .  This approximation is called the method of exhaustion. It was developed used by Archimedes.

Note: If you do not see the applet above, download and install Java

Tuesday, July 17, 2012

GeoGebra External Tutorials

GeoGebra Tutorials

Monday, July 16, 2012

Miscellaneous GeoGebra Applets

Miscellaneous GeoGebra Applets

Monday, July 9, 2012

Tutorial 5 - Graphs and Sliders

Objective: To explore the effects of a and b in the graph f(x) = a sin(x) + b.
Open GeoGebra and use the View menu to display the coordinate axes.
2slider.pngSelect the Slider tool and click on the Graphics view to display the Slider dialog box.
In the slider dialog box, select Number from the option buttons, and leave the other options as is. GeoGebra will automatically givea as the slider name.

Now create slider b similar to slider a.
Type f(x) = a*sin (x) + b in the input bar and press the ENTER key to graph it.
Move the small circles on both sliders. What do you observe?
< Previous Tutorial

Tutorial 4 - Graph, Text, and Latex

Objective: To construct graphs of linear and quadratic functions.
Tools: Insert Text

Open GeoGebra. Use the View menu to display the coordinate axes.
Type y = - x + 4 to construct a line passing through A and B.
Type f(x) = 0.5x^2 - 4.5x + 9 to construct a parabola passing through points A and B.
To construct the intersection, type A = (2,2) and B = (5,-1) in the input bar and press the ENTER key after each equation.
5text.pngWe label the graphs. To label the graph of the quadratic function, click the Insert Text tool, and click on the Graphics view near the graph of the quadratic function to display the Text dialog box.
In the text dialog box, type f(x) = 0.5x^2 - 4.5x + 9 in the Edit text box, then click the Latex formula check box to check it, and then click the OK button. Notice what happens to the text.
Now, label the graph of the linear function and the coordinates of A and B with (2,2) and (5,-1) respectively.
< Previous Tutorial ** Next Tutorial >

Tutorial 3 - Angles and Object Properties

Objective: To use the Angle tool, Styling Bar and Object Properties.Tools: Angle


Open the Tutorial 2 file. Use the View menu to hide the Algebra view.
2angle.pngTo measure angle BAC, select the Angle tool, and click points BAC in that order. Notice that an angle symbol appears on angle A. Use the same tool to measure the remaining interior angles. For the mean time, ignore the reflex angles.
Click the angles and change their colors on the Styling Bar located at the upper-left of the Graphics view. Apply similar colors to congruent angles.
To disallow reflex angles, right click on angle A* (right click the sector), and select Object Properties from the pop-up menu.
In the Object Properties window, click the Basic tab, and uncheck the Allow Reflex Angle check box, and click the Close button. Repeat the process to disallow reflex angles on other interior angles.

< Previous Tutorial ** Next Tutorial >

Tutorial 2 - Lines, Circles, and Angles

Objective: To constructt a rhombus and display the measure of its interior angles.Tools: Compass, Parallel Line, Angle

1segment.pngConstruct segment AB.
2compass.pngSelect the Compass tool, click on the segment and then click on A. This will form a circle with center A and passing through B.
3point.pngConstruct point C on the circle.
4segment.pngConstruct segment BC. Move point C, what do you observe?
Right click on the circle and then click Show Object from the pop-up menu to hide it.
6parallel.pngTo construct a line parallel to AB passing through C, select the Parallel line tool, click on segment AB, and then click on point C. Now, use the same tool to construct a line parallel to AC and passing through B.
7intersect.pngNow, intersect the two lines to form the fourth vertex of the rhombus. Hide the two lines and construct segments BD and CD.
8segment.pngHide the two lines by right clicking them and clicking Show Object, and then construct segments BD and CD.
< Previous Tutorial ** Next Tutorial >

Tutorial 1 - Points, Segments, and Intersections

Objective: To construct a triangle and find its centroid.Tools: Move, New Point, Segment between Two Points, Midpoint or Center, Intersect Two Objects

NoTool    Instruction
Select the New Point tool and click three different locations on the Graphics view.
2segment.pngTo construct side AB, select the Segment between Two Points tool click point A, and then click point B.
Use the same tool to construct segments AC and BC .
3move.pngSelect the Move tool and drag the points. What do you observe?
4midpoint.pngTo construct the midpoint of AB, select the Midpoint or Center tool, click point A, and then click point B. Now, get the midpoints of AC and BC. After this step, your drawing should look like the following figure.

5segment.pngTo construct the medians, use the Segment between Two Points tool to construct segments AFBE, and CD, the medians of the triangle.
6intersect.pngTo intersect the medians, select the Midpoint or Center tool, and then click any of the two medians.

Explore More!

  • Move points AB, and C. What do you observe about the medians?
  • What conjecture can you make about the medians of a triangle?
  • Move the points and the segments on the figure. Which objects can/can't be moved? Why?

Next Tutorial >

The GeoGebra Toolbar

The GeoGebra toolbar is used for constructing and exploring geometric objects. The tools are categorized into twelve as shown below. In each category, the default tool is displayed. For example, in the Special Line tools, the Perpendicular line is the default tool.

More tools in each category can be displayed by clicking the triangle at the bottom-right of the toolbar buttons. Once the list of tools is displayed, the user can select a tool by clicking it. The icon of the selected tool will then be displayed on the toolbar.

During construction, the selected tool is highlighted by the blue border as shown in both figures. To deselect a tool, you can click the Move tool or press the Esc key on your keyboard.

The GeoGebra Interface

The GeoGebra window as shown below is capable of simultaneously displaying algebraic, geometric, and numeric representations of mathematical objects. A function, for example, can be viewed as a graph, an equation, and a table.
external image parts-of-geogebra.png

Parts of GeoGebra
  • Menu bar - used for managing files, modifying preferences, and customizing settings.
  • Toolbar - used for constructing and modifying mathematical objects.
  • Graphics view - used for displaying geometric representations of mathematical objects
  • Algebra view - used for displaying algebraic representations of mathematical objects.
  • Input bar - used for inputting equations and performing mathematical computations

Saturday, July 7, 2012

What is GeoGebra?

GeoGebra is a dynamic mathematics software that joins algebra, geometry statistics, calculus, table and graphing in one easy-to-use package. GeoGebra can be used to construct mathematical objects such as points, vectors, segments, lines, polygons, conic sections, and functions, geometrically (using the mouse) or algebraically (using the keyboard). These objects can be entered and modified directly on screen or through the command line.

Here are some features of GeoGebra:

  • Graphics, algebra, and tables are connected and fully dynamic
  • Easy to use interface, yet many powerful features
  • Authoring tool to create interactive learning materials such as web pages
  • Available in many languages
  • Free, open-source, and runs in multiple platforms

GeoGebra was created by Markus Hohewarter in 2001, and is now continually developed by a team of international programmers.

GeoGebra Institute Trainers

The following are the GIMM's trainers:

  • Guillermo P. Bautista Jr.
  • Erlina R. Ronda
  • Allan M. Canonigo

Tuesday, July 3, 2012

Intermediate GeoGebra Tutorials

GeoGebra Tutorials - Intermediate Leve

Sunday, July 1, 2012


Official Web Page



Games and Puzzles