GeoGebra Basic Construction 5– Rectangle

In this GeoGebra tutorial, we use the Perpendicular Line and Parallel Line tools to construct a rectangle. The idea is to construct segment AB, construct two lines perpendicular to it, one passing through segment A and the other through segment B. Next, we will construct point on the line passing through B, then construct a line parallel to AB passing through C. The fourth intersection will be our point D. If you want to follow this construction step-by-step, click here to open GeoGebra on your browser.

Step-by-Step Construction

1. Open GeoGebra and select Geometry from the Perspective menu on the side bar.
2. To automatically show the labels of points and not the other objects, click the Options menu, click Labeling, then click New Points Only.
3. To construct a rectangle, select the Segment between Two Points tool and click two distinct places on the Graphics view to construct segment AB.
4. We construct two lines that are both perpendicular to AB, one passing through A and the other through B. To do this, click the Perpendicular Line tool, click point A and click the segment. To construct another line, click point B then click the segment. After these steps, your drawing should look like the one shown in Figure 1.

Figure 1

5. Next, we construct a point on the line passing through point B. To do this, click the New Point tool, and then click on the line passing through B.
6. Now, to create a line parallel to AB, passing through point C, click the Parallel line tool, click on point C and then click on the segment.

Figure 2

Your drawing should look like the one shown in Figure 2 after this step.

7. Using the intersect tool, we construct the intersection of the line passing C (parallel to AB) and the line passing through A. To do this, click the Intersect Two Objects tool and click the two lines mentioned.
8. Now, we hide the three lines. To do this, click the Show/Hide Object tool and click the three lines. Notice that the lines are highlighted. Click Move tool to hide the lines.
9. Use the Segment between Two Points tool to connect the points to construct the rectangle.
10.  Drag the points of the vertices of the rectangle. What do you observe?

GeoGebra Tutorial 16 – Slider, Sequence and Segment Division

This is the 16th GeoGebra Tutorial in the GeoGebra Tutorial Series. If this is your first time to use GeoGebra, I suggest that you read the GeoGebra Essentials Series first.

In this tutorial, we use the Slider tool and the sequence command to divide a  segment into n equal partitions. If you want to view the final output of the tutorial click here.

Step-by-step Instructions

1. Open GeoGebra and select the Algebra & Graphics view from the Perspectives menu.
2. To label points only, click the Options menu, click Labeling, and then click New Points Only.
3. To create point A on the origin, select the Intersect Two Objects tool, click on the x-axis then click on the y-axis.
4. To create point B on the x-axis, select the New Point tool and then click anywhere on the positive x-axis.  Drag point B to be sure that it moves only along the x-axis. If not, delete the point and repeat this step.
5. Next, we construct ray AB. To do this, click the Ray through Two Points, click point A, and then click point B.
6. Next, hide by clicking on its icon on the upper left of the Graphics view.

Figure 1

7. Next, we hide by right clicking on point B, and then clicking on Show Object in the context menu.

8. Now, we create a point C on the ray. The idea of steps 3-7 is to ensure point C will only move along the ray and only on the positive x-axis. This is because we are going to use the point at the origin as the leftmost point of the segment AC.
9. Now, we hide the ray. To do this, right click the ray (not the points) and uncheck Show Object by clicking it. That only leaves points A and C in the Graphics view.
10.  Select the Segment through Two Points tool, click point A and click point C to construct segment AC.  Your drawing after this step should look like the figure below.

Figure 2

11.  Our task is to divide segment into AC into n parts.  So, we will create a slider with value from 1 to 20, and an increase of 1. To do this, select the Slider tool and click anywhere on the Graphics view to display the Slider dialog box (see Figure 3).

12.  In the Slider dialog box choose Number, change the name to n, the minimum value to 1, maximum value to 20 and increment to 1 (see Figure 3), then click the Apply button.

13.  Next, we compute for the partition length of the segment if it is divided by n partitions. To do this, we divide the length of the segment by n.  If you have followed the tutorial correctly, the length of the segment should be represented by b in the Algebra view. But to be sure, let us use the distance command to get the distance between A and C. We use d to denote the distance between A and C. To do this, type d = distance[A,C].
14.  Move point C and observe what happens to the value of d in the Algebra view.
15.  We now compute for the partition length p. Type p = d/n, then press the ENTER key on your keyboard.
16.  Next, we use the sequence command to produce points that will serve as division markers. To do this, type the following command and then press the ENTER key.

Sequence [(p*i, 0), i, 1, n]

The ordered pair (p*i,0) is the coordinate of the points that divide the segment.  Note that i is a series of numbers running from 1 through n. For more information about the sequence command, see the explanation below.

17.  Now, move the slider to 20. Your drawing should look like the figure below.

Figure 4

18.  Move point C back and forth, vary the value of n, and observe what happens.
19.  You can change the value of n in the Slider dialog box. To display the slider dialog box, right click the slider and click Object Properties.

Explanation of the Sequence Command

The sequence command is just like the summation symbol. Their only difference is that the sequence command is that it only generates its terms and does not add them.

The general forms of the sequence command are

Sequence[<expression>,< variable>, <start value>,<end value>]

Sequence[<expression>,< variable>, <start value>,<end value>, <increment>]

For example, if we want to plot points with coordinates (1,1), (2,2), (3,3), (4,4), and (5,5), then the sequence command would be sequence [(i,i), i,1, 5]. The letter i is our dummy variable, 1 is the start value and 5 is the end value. By default, the increment of the sequence command is 1 unless stated otherwise (see general form).

In the command in step 16, we want construct points with coordinates (p*i, 0) to the right of A with coordinates (0,0).

Sequence[(p*i, 0), i, 1, n]

To give you a more concrete explanation, suppose the length of segment AC is 5 units and we want to divide it to 20 partitions. Hence, d = 5 (recall that d is the distance between A and C, see step 13) and n = 20. In step 15, we computed for the partition length p = d/n = 20/5 = 0.4. From the given n=20, the coordinates of our points is shown below.

Figure 5

The rightmost column in the table shows the coordinates of the 20 points generated by the sequence.

GeoGebra Tutorial 15 – Circle Area Approximation and Circumscribed Polygons

This is the continuation of GeoGebra Tutorial 14 – Circle Area Approximation and Inscribed Polygons.  You must finish the said tutorial before doing this because you will use the output file. If you want to follow this tutorial step-by-step, you have to download first the output file of the previous tutorial (go to the output file, click File>Save).

If you want to see the final output of this tutorial, click here.

Step-by-Step Instructions

1. Set the sliders to n = 3 and r = 3 to construct a circle with radius 3 and an inscribed triangle.
2. To construct a line tangent to the point B at (3,0), select the Tangents tool, click point B and click the circumference of the circle.

3. Using the steps in 2, create two more tangents, passing through the two vertices of the triangle; that is, with the Tangent tool selected, click the circle and the point on the circle. After step 3, your drawing should look like the Figure 1.
4. Select the Intersect two objects tool, and click the two lines intersecting at the first quadrant, then click the two lines intersecting at (-6,0) to intersect them.

Figure 1

5. Now, right click  one of the tangent lines and click Show Object from the context menu to hide it.  Hide the other two tangent lines.
6. We now create a circumscribed polygon using the Regular Polygon tool. To do this, click the Regular Polygon tool, and click the first intersection and click the second intersections of tangents to reveal the Regular Polygon dialog box.
7. In the Points text box, type n.This means, that whatever the polygon that is inscribed in the circle, that also will be the polygon that will circumscribe it.
8. Use the text tool to display the area of the circumscribed polygon as shown in Figure 2.

Figure 2

9. Adjust the sliders n = 3 and r = 1. What is the area of the circle?
10.  If you want to change the maximum number of sides, right click Slider n and click Properties. This will display the Slider dialog box.
11.  In the Properties window, click  the Slider tab, change the maximum value to 100, then click the Close button.

12.  Move the sliders and observe the relationship of the areas of the three objects. What conjectures can you make from your observations?
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