GeoGebra Tutorial 30 – Loci, Roses and Reflections

This is the 30th GeoGebra Tutorial of 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 reflect a point on a unit circle about the x-axis and about the y-axis to form rectangle ABCD. We construct a segment with length equal to the area of the rectangle passing through point B.  One of the endpoints of this segment is at the origin.  As we move the point B along the the circle, the trace of the end point of the segment that not on the circle’s center will form a 4-petal rose.

If you want to follow this tutorial step-by-step, you can open the GeoGebra window in your browser by clicking here.  You can view the output of this tutorial here.

Step-by-Step Instructions

1. Open GeoGebra and select Algebra & Graphics from the Perspectives menu.
2. To show labels of new constructed points only, click the Options menu, click Labeling, then click New Points Only.
3. We create a circle with center (0,0) and radius 1. To construct point A, the center of the circle, select the Intersect Two Objects tool, click the x-axis, then click the y-axis.
4. To construct the circle with center A and radius 1, type circle[A,1] and press the ENTER key on your keyboard.
5. Next we construct a point on the circle and color it red. To construct point B on the circle, select the New Point tool and click the circumference of the circle. Be sure that point B is on the circle. If not, delete it and repeat the construction.
6. We now reflect point B about the y-axis and x-axis. To do this, select the Reflect Object About Line tool, select point B and then click the x-axis. This will produce B’. Your drawing should look like Figure 1.

Figure 1

7. Now we reflect B about the y-axis. To do this, with the Reflect Object About Line tool still active, click point B and click the y-axis. Notice that the label of the B’ changed to B’1 and the newly produced point became B’.

Figure 2

8. Now, reflect B’ about the y-axis to construct the fourth vertex of the rectangle.
9. Now, we change the color of point B to red or any color not blue.
10.  We now rename the blue-colored points. Right click each point, select Rename from the context menu and change the names of the blue points to C, D, E as shown in Figure 3.
12.  Next, we use the Polygon tool to construct rectangle BCDE. Select the Polygon tool, and click in the following order: B, C, D, E and B to close the polygon. Note that the rectangle’s name is poly1 (see Algebra view).
13.  Next, we will construct circle with radius poly1. The value of poly1 in the Algebra view is the area of the rectangle. To do this, type circle[A,poly1] and press the ENTER key on your keyboard.
14.  Select the Ray through Two Points tool, select point A and select point B to construct ray AB.

15.  Next, using the Intersect Two Objects tool, intersect Ray AB and the new constructed circle This will produce point F. This means that the length of AF is equal to the area of the rectangle.

Figure 3

16.  Next, we hide the ray and the circle containing point F. To do this, right click each object and click Show Object from the context menu.
17.  Connect AF using the Segment Between Two Points tool.
18.  We now trace point F. To do this, right click point F and select Trace On from the context menu.  Now, move point B along the circle. The trace should look like Figure 3.

Figure 5

19.  Now, to create a solid curve tracing the Select the Locus tool, select point B and select point F. Now, your drawing should look like Figure 5.
20.  Now, use the Move tool and move the drawing pad.
21.  Hide the names of the points by right-clicking each point and selecting Show label.

Figure 6

The shape that we have created is called the four-leaf petal rose, a sinusoid plotted in the polar coordinates.

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