The Parallel Postulate

In the figure below, AB and CD are parallel lines. EF is a transversal.

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 parallel postulate states that given two parallel lines (on the same plane) and a transversal, their corresponding angles are congruent. Using this postulate as shown above, determine the measure of the other angles and record your observations. Make a conjecture about them.

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GeoGebra Graphics 2 and the Sine and Cosine Functions

Move the point on the circle in the left window (Graphics 1).

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

This applet demonstrates how to use Graphics 2 of GeoGebra. Graphics 1 shows the distances on the circle: distance traveled (a)  by a point about the circle from (1,0), it's distance from the x-axis (b), and its distance from the y-axis (c).

Graphics 2 shows the two points with coordinates (a,b) and (a,c). The traces of the points in Graphics 2 forms the graph of the sine function and the consine function from 0 to pi.


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The Mean Proportion

Move point Q and observe what happens.

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

In the figure, PSR is a semi-circle and with PR as the diameter.  Given a rectangle with side QR, and a triangle PSR, a square is formed using its height. The  square has the same are as the rectangle.

Download GGB file here.

GeoGebra 4.0 Tutorials and more...

This is to inform everyone that I am currently updating my GeoGebra Tutorials in Mathematics and Multimedia. I have updated the following tutorials so that they are compatible to GeoGebra 4.0.

• Introduction to GeoGebra
• GeoGebra Essentials 1 – Locating the Triangle’s Centroid
• GeoGebra Essentials 2 – Constructing a Rectangle

I will be updating three or more tutorials this weekend. If you are new to GeoGebra, I have created more than 50 GeoGebra tutorials categorized as follows.

• GeoGebra Essentials Series (10 tutorials for beginners)
• GeoGebra Basic Geometric Construction Series (7 tutorials for beginners)
• GeoGebra Intermediate Series (30 tutorials for beginners and intermediate users)
• GeoGebra Advanced (10 tutorials for advanced users)
• GeoGebra 4.0 Sneak Peek (New!) – 10 posts about GeoGebra 4.0

Please share the tutorials to people who might be interested. 

Polygons and Tessellation

Move the slider and observe what happens.

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

1. Which of the regular polygons tessellate? Can you explain why they do?
2. As n increases can we can find a regular polygons that tessellate? 
• If yes, how many of them can be found? Are they finite?
• If no, what is the maximum number of side of a polygon that can tessellate? Why?
3. Make a conjecture about your observations above. 

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If you want to know more about tessellation, you may want to check out the following articles that I have written:

• Tessellation: The Mathematics of Tiling
• Regular Tessellations: Why only three of them?

van Aubel's Theorem

Squares are placed outwardly at each side of quadrilateral ABCD. The center of the squares on the opposite sides are connected with segments (PN and MO). Move points A, B, C and D and observe what happens. Make a conjecture about your observations.

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

Van Aubel's Theorem

Given an arbitrary planar quadrilateral, place a square outwardly on each side, and connect the centers of opposite squares. Then van Aubel's theorem states that the two lines are of equal length and cross at a right angle.

Click here to download GGB file.

GeoGebra 4.0 offline installer, now available

If your PC/Mac is not always connected to the internet, or you have a computer laboratory with no internet connection, you can now download the GeoGebra 4.0 offline installer.

GeoGebra 5.0 Beta is also in development. 

Area of a Trapezoid

Move point A to change the shape of the trapezoid.  Then move slider  to the extreme right to rotate the trapezoid.

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

1.) What kind of quadrilateral  is formed when  equals 180 degrees? Justify your answer.
2.) What is the relationship between the area of trapezoid ABCD and the area of the quadrilateral formed in 1?
3.) Using the quadrilateral formed in 1, find the area of the trapezoid. Hint: Use  and .


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Testing Post - Social Network Integration

Recently, GeoGebra Central's automatic feed to Facebook and Twitter stopped. I am currently troubleshooting so please ignore this post.