- Mathematics and Multimedia

This is the seventh tutorial of the GeoGebra Intermediate Tutorial Series. If this is your first time to use GeoGebra, I strongly suggest that you read the GeoGebra Essentials Series.

In the Graphs and Sliders posts (click here and here to view posts), we have discussed how to use sliders as numbers. In this tutorial, we use the slider control to determine the measure of angles, and to rotate a triangle to show that the sum of its interior angles is 180 degrees. This is the same GeoGebra worksheet shown in my Parallel Lines and Transversals post, but we will change some of the labels. Although this tutorial is the seventh of the GeoGebra Tutorial Series, it is independent from the other tutorials, so you can follow it step-by-step without having to learn first the previous six tutorials.

Figure 1 - Rotated triangels using sliders.

Construction Overview

Our construction will start by drawing line AB and constructing triangle ABC using the polygon tool. Afterwards, we will reveal the interior angle measures of the triangle and create two angle sliders namely $\alpha$ and $\beta$ . Next, we rotate the triangle 180 clockwise about the midpoint of BC producing triangle A’B’C’ (see Figure 1-B). We then repeat the process, and rotate triangle A’B’C’ 180 degrees clockwise about the midpoint of A’C’ to produce A’’B’’C’’.

Part I – Construction Triangle ABC

	1.) Open GeoGebra. Hide the Algebra view and the Coordinate axes (View menu).
	2.) Click the Line through Two Points tool, and click two distinct places on the Graphics view to construct line AB.
	3.) If the labels of the points are not displayed, click the Move tool, right click each point and click Show label from the context menu. (The context menu is the pop-up menu that appears when you right click an object).
	4.) Click the New Point tool and construct a point C not on line AB.
	5.) Display the name of the third point. GeoGebra would automatically name it C, otherwise right click and rename it C. Note that it is important that you follow the labels because we are going to use the labels later to refer to objects.
	6.) Click the Polygon tool and click the points in the following order: point A, point B, point C, and click again point A to close the polygon. Your drawing should look like the drawing in Figure 1. Figure 2 - Triangle ABC on line AB.
	7.) Move the vertices of the polygon. What do you observe?
	8.) Now we construct two angle sliders $\alpha$ and $\beta$ . To do this, click the Slider tool, and click the drawing area.
	9.) In the Slider dialog box (see Figure 3), choose the Angle radio button, and then leave the name angle as $\alpha$ . In the Interval tab, choose 0° as minimum, 180° as maximum and 1°, and then click the Apply button when finished. Figure 3 - The Slider dialog box
	10.) Using steps 8-9, create another slider with the same slider specifications shown in Figure 3 and name it $\beta$ . You can find the Greek letters by pressing the $\alpha$ button located at the right of the text box.
	11. ) We reveal the angle symbols for all the interior angles of a triangle. We will change the colors of the angle symbols, angle A to red, angle B to blue and angle C to green without revealing their name and their actual measures. To do this, click the Angle tool and then click the interior of triangle ABC.
	12.) To hide the labels of the angle symbols, right click each angle symbol and uncheck Show label from the pop-up menu.
	13. To change the color of the angle symbol of angle A, right click the angle symbol (not point A) and click Properties.
	14. In the Properties window, click the Color tab and choose the color you want from the color palette then click the Close button.
	15. Change the color of angle B to blue and leave angle C as is. Your drawing should look line Figure 1-A after step 15.

Part II – Rotating the Triangle

We already have the sliders ready. The next thing that we will do is to rotate the triangle. The idea is to create a rotation point. Our choice would be the midpoint of BC. That is because if we rotate ABC by 180 degrees producing A’B’C’, angle A’C’B’ will be adjacent to angle ABC (see Figure 1-B). This is also the idea when we rotate A’B’C’ producing A’’B’’C’’.

	1.) To get the midpoint of BC, click the Midpoint or Center tool, and click side BC (the segment, not the points).
	2.) Note that we want ABC to rotate around the midpoint of BC, whatever the measure of our slider P. To do this, choose Rotate around a Point by Angle tool, click the interior of the triangle and click the midpoint of BC to reveal the Rotate Object dialog box.
	3.) In the Rotate Object dialog box, change the measure of the angle to $\alpha$ , choose the clockwise radio button, and then click the OK button. Figure 4 - The Rotate Dialog Box
	4.) Now move slider $\alpha$ . What do you observe?
	5.) Adjust slider $\alpha$ to 90 degrees, and show the labels of the vertices of the rotated triangle. (Refer to Part I – Step 3).
	6.) While the triangle is still rotated 90 degrees, click the Angle tool and click the interior of triangle A’B’C’. Hide the labels of the angles symbols.
	7.) Change the colors of the angle measures. Refer to Part I – Steps 13 through 15. Be sure that angle A and A’ have the same color, B and B’ have the same color, and C and C’ have the same color. Your drawing should look like the drawing in Figure 6. Figure 5 - The Rotated Triangle

Part III – Creating the Third Triangle

The idea of creating the third triangle is basically the same as that of creating the second triangle, so I will just enumerate the steps and left the construction as an exercise.

Get the midpoint of A’C’. (Refer to Part II Step 1)
Rotate triangle A’B’C’ $\beta$ degrees clockwise around the midpoint of A’C’. (Refer to Part II – Steps 2 – 3).
Reveal the labels of the vertices of the third triangle which is A’’B’’C’’. (Refer to Part II – Step 5 and Part I – Step 3).
Reveal the angle symbols of triangle A’’B’’C’’. (Refer to Part I – Step 11)
Hide the labels of the angle symbols, and change the colors of the angle symbols of triangle A’’B’’C’’. (Refer to Part I – Step 13-15)