## GeoGebra Tutorial 10 – Vectors and Tessellation

This is the 10th tutorial in the GeoGebra Intermediate Tutorial Series. If this is your first time to use GeoGebra, you might want to read first the GeoGebra Essentials Series.

This tutorial is the sequel of GeoGebra Tutorial 9 – Vector and Translation.  In this tutorial, we  use the idea of translation to tessellate  the plane. Tessellation is a process of covering a plane with no gaps and no overlaps.

The final output of our tutorial is shown in Figure 1 and the GeoGebra applet can be viewed here.

Figure 1 – Octagon and squares tessellating the plane.

To give you the whole picture, I have enumerated the summary of what we are going to do in this construction.

1. Construct an octagon containing points A at (0,5) and B at (0,4).
2. Draw point O at the origin which is the initial point of vector, P at the positive x-axis and vector Q at the negative y-axis. P and Q are the terminal points of the vectors.
3. Draw vectors u (containing O and P) and v (containing O and Q).
4. Translate the octagon to tessellate the plane using vector u.
5. Translate all the created octagons down using vector v.
6. Draw a square that will cover the space at the center of the 4 leftmost adjacent octagons.
7. Translate the square to the right using vector u.

Construction Protocol

 1.) Open GeoGebra and select Algebra & Graphics from the Perspectives menu. 2.) Click the New Point tool and place two points in the coordinates given: A on (0,5) and B in (0,4). 3.) Select the Regular Polygon tool, click point A, then click point B to display the Regular Polygon dialog box. In the dialog box, type 8, then press the OK button. If the labels of the points and the segments are displayed, remove them by right clicking them and unchecking on Show label from the context menu. Figure 2 – Octagon with containing segment AB. 4.) Next, we create three points which will be the initial and terminal points of the vector (see Figure 1). Click the New Point tool, click on the origin, click a point on the positive x-axis near the origin and the negative y-axis near the origin. 5.) We now rename the three points. To rename the point on the origin, right click it and click Rename from the context menu. In the Rename box, type O and press the OK button. Rename the point on the x-axis as P and the point on the y-axis Q. 6.) Now, to create vector u, select the Vector between Two Points tool, click on point O, then click on point P. To create vector v, with the Vector between Two Points tool still active, click point O and then click point Q to create vector v. 7.) To translate the octagon to the right, select the Translate Object by Vector tool, click the interior of the octagon, then click vector u. 8.) Adjust vector u such that the right vertical side of the first octagon coincides with the left vertical side of the translated octagon. Your drawing should look like Figure 3. Figure 3 – An octagon and its translation using vector u. 9.) To translate another octagon, with the Translate Object by Vector tool still active, click the rightmost octagon, then click vector u. Repeat this step three times giving us 5 octagons with adjacent vertical sides. 10.)  Next, we translate the octagons down vertically. To do this, select the Translate Object by Vector tool, click the leftmost octagon and then click vector v. Adjust the translated octagon by moving point Q such that the lower horizontal side of the original octagon coincides with the upper horizontal side of the translated octagon. 11.)  Repeat step 10 until all the 5 octagons are translated down.  After step 11, your figure should look like Figure 1 with white squares. 12.)  The last part of the task is to cover the empty square spaces between octagons. To do this, click the Regular Polygon tool, click the two consecutive points of the leftmost square (on the side of the octagon) to display the Regular Polygon dialog box. 13.)  Type 4 as the number of Vertices and then click OK. If the square created is on the other side, undo the step, and reverse the order of the click. 14. ) To tessellate, click the square and then click vector u. Repeat the translation as you have done in step 9.

## GeoGebra Basic Construction 3 – Right Triangle

In this construction, we  use the Perpendicular Line tool to create right triangle ABC where angle B is the right. First, we construct segment AB, then construct a line perpendicular to segment AB and passing through B. Then, we construct point C on the line, hide the line and connect B to C as well as connect A to C with the segment tool.

Figure 1 – Triangle ABC right angled at B.

The detailed steps are enumerated below.

 1.) Open GeoGebra and select Geometry for the Perspectives menu at the Sidebar. 2.) Click the Segment between Two Points tool and click two distinct places on the Graphics view to construct segment AB. 3.) If the labels of the points are not displayed, click the Move button, right click each point and click Show label from the context menu. 4.) Next, we construct a line parallel perpendicular to segment AB and passing through point B. To do this, select the Perpendicular Line tool, click segment AB, then click point B. 5.) Next, we create point C on the line. To do this, click the New Point tool and click on the line. Your drawing should look like the figure below. Display the label of the point in case it is not shown (see no. 3) Figure 2 – Line BC perpendicular to segment AB. Bbe sure that you can only drag point C on the line. Otherwise, you have to delete the point and create a new point C. 8.) Next, we hide the line passing through B. To hide the line, right click the line and uncheck Show Object from the context menu. 9.) Select the Segment between Two Points tool and connect B and C. With the same segment tool, connect A and C. 10.) Using the Move tool, drag the vertices of the triangle. What do you observe? 11.) If you want,  can use the Angle tool to verify the measure of angle B. To do this, click the Angle tool, and click the  vertices of the triangle in the following order: point C, point B and point A.

You can also construct and equilateral using other tools. As an exercise, try constructing a right triangle using the following:

1. Circle tool and segment tool
2. Angle Bisector Tool

## GeoGebra Tutorial 9 – Vector and Translation

This is the ninth tutorial in the GeoGebra Intermediate Tutorial Series. If this is your first time to use GeoGebra, you might want to read the GeoGebra Essentials Series.

In this tutorial, we use the Vector between Two Points tool to translate a triangle and investigate the relationship between its preimage and image. We will also use the grid in this tutorial.

If you want to follow this tutorial step-by-step, you may open the GeoGebra window in your browser by clicking here.

 1.) Open GeoGebra and select the Algebra & Graphics view from the Perspectives menu. 2.) Display the grid by clicking the View menu and choosing Grid. 3.) Click the New Point tool and place the points on the coordinates given: A on (2,3), B on (4,1) and C on (5,2). 4.) Next, we draw triangle ABC using the Polygon tool. To do this, click the Polygon tool and click the points in the following order: point A, point B, point C and point A again to close the polygon. 5.) To display the label and the coordinates of the points, right click the points then click Object Properties to display the Preferences dialog box. 6.) In the Basic tab of the Preferences dialog box, check the Show label check box, and choose the Name & Value option in the drop-down list box, and then close the window. Your drawing should look like the figure below. Figure 1 7.) The only remaining part of the construction is the vector tool. To construct vector DE, select the Vector between Two Points tool, click the origin and click the coordinate (1,2). After this step, your drawing should look like the one shown in Figure 2. Figure 2 8.) To translate the object using the vector, select the Translate Object by Vector tool, click the triangle and then click the vector. Notice that a translated triangle appears after clicking the vector tool. 9.) What can you say about the preimage of the triangle object and the translated object? 10.) If the coordinates of the vertices of the translated triangle is not displayed, display it using the steps we have done in step 5 and 6. 11.) What do you observe about the relationship of the coordinates of the points of the original triangle and the translated triangle? 12. Move the terminal point (point E) of the vector. Does your observation in (11) still hold?
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