## Dissecting an Octagon to Two Smaller Octagons

To dissect the octagon, move slider alpha to the extreme right and then move slider v to the extreme right.
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:

Contributed by: Izidor Hafner
Based on work by: Greg N. Frederickson

Snapshot

## Multiplication Table

Move the Column and Row sliders to choose the numbers you want to multiply.

## GeoGebra Tutorial: How to change the x-interval to pi or pi/2

In graphing Trigonometric Functions, there are cases that you need to change the interval of the x-axis in terms of π.  In this tutorial, you will learn how to change the interval of the x-axis.

1.) Right click a blank space on the Graphics view and then click on Graphics… from the pop-up menu to display the Preferences dialog box.

2.) In the Preferences dialog box, click on the xAxis tab, and then check the Distance check box.

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3.) In the drop-down list box at the right of the Distance check box, select π or π/2 and then click the Close button to close the dialog box.

After step 3, the interval of the x-axis should be π or π/2.

## Cross to Isosceles Triangle Dissection

Move the Size slider to change the size of the cross and move the slider $\alpha$ to the extreme right to transform the figure.

The applet above is an example of a dissection puzzle. A dissection puzzle, also called a transformation puzzle tiling puzzle where a set of pieces can be assembled in different ways to produce two or more distinct geometric shapes (Wikipedia).

The cross to isosceles triangle dissection puzzle above was created by Henry Dudeney in 1897.

## Folding an Ellipse

Paper Folding Activity

1. Draw a circle and draw a point in its interior.
2. Mark a point anywhere on the circle.
3. Fold such that the two points coincide and crease well.
4. Repeat steps 2-3 as many times as you can.

What do you observer about the creases?Note: The green lines represents the folds on the circle.

## GeoGebra Tutorial: Constructing a Cardioid

A cardioid is a curve formed by moving circles. One of the ways to form a cardioid is shown below. In the applet (move the slider to the extreme right), the cardioid is given as the envelope of circles whose centers lie on a given circle and which pass through a fixed point on the given circle.

Step by Step Instruction in Constructing the Applet Above

1. Construct point A at (0,0) and point B at (0,1).
2. Construct a circle with center A and passing through B
3. Construct an angle slider and set the minimum number to 0°, maximum number to 360° degrees and increment of 2°.
4. Construct angle BAB‘. To do this, select the Angle with Given Size tool, click on point B and click on point A to display the Angle with Given Size dialog box.
5. In the Angle with Given Size dialog box, replace 45° with α,  choose counterclockwise button, and then click OK.
6. Hide the green sector by right clicking it and then clicking on the Show Object from the pop up menu.
7. Now construct a circle with center B‘ and passing through B.
8. Right click the circle with center B and  click Trace on.
9. Move the slider and observe what happens.

## GeoGebra Tutorial: How to Disable Reflex Angles

In GeoGebra, we use the Angle tool to reveal angle measures.  By default, an angle can measure  up to 360°; howeverthere are instances that we want it to be less than or equal to 180°.  For example, we need to limit the angle measures of the interior angles of a polygon to 180° if we want to find its angle sum.

In this tutorial, we learn how to disable angles measuring more than 180° but less than 360°.  Such angles are called reflex angles.  To disable reflex angles, do the following:

1. Right click the angle measure (green sector) and click Object Properties to display the Preferences window.

2. In the Preferenceswindow, click the Basic tab.
3. Select 0° and 180° in the Angle Between dialog box and then close the window.

## Polygon Names

Move the slider to change the number of sides of the polygon.   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 is designed for elementary school pupils to help them memorize the names of polygons.

## The Inscribed Angle Theorem

In the figure below, angle BAC is an inscribed angle and angle BOC is the central angle of circle O. Both angles are subtended by the same arc. Make a conjecture about these two angles.
Check the Show/Hide Angle Measures check box and move any of the points. Is your conjecture always correct? Prove your conjecture.

## GeoGebra Tutorial: Using the Perpendicular Line and the Point on Object Tools

In this tutorial, we use GeoGebra to explore the minimum sum of the distances of a point on the interior of a triangle to its sides. We learn how to use the Point on Object tool and the Perpendicular Line tool. We also learn how to compute using the Input bar. The final output of this tutorial is shown in the following applet.
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
Steps in Creating the Applet

 No Instruction 1 Using the Polygon tool, construct triangle ABC. 2 Select the Point on Object tool, and then click the interior of the triangle to create point D. 3 Move point D and observe what happens. Can you move point D outside the triangle? 4 Next, we create a line perpendicular to AB passing through D.  To do this, select the Perpendicular Line tool, select segment AB, and then select point D. 5 Create two more lines passing through D, one perpendicular to AC and the other perpendicular to BC. 6 Using the Intersect Two Objects tool, intersect the lines and the perpendicular segments. Your drawing should look like the figure below. 7 Hide the three lines by right clicking each line and clicking them and selecting Show Objects. 8 Using the Segment between Two Points tool, construct segments DE, DG, and DF. 9 To find the total distance t of points D from E, F, and G, type t = g + h + i, and then press the ENTER key. 10 Move the points to explore and observe the value of t.

Application

A building is going to be constructed with a triangle bounded by the roads connecting Buildings A, B, and C. A connecting pathway is to be constructed from the three roads to the building. Find the location of the building that will minimize the cost of pathway construction.

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