Example 3. Constructing an equilateral triangle (with an introduction to the compass and hiding command).
It is important that you remember the properties that each geometric shape has. This is important to know as you construct each shape. In the case of an equilateral triangle, recall that it is a triangle with three congruent sides. (Refer to Construction 10 of Chapter 14 in your textbook to see how an equilateral triangle is constructed.) Those directions are the same for the software. Remember that colors are only provided in version 4 so when we say "a colored" object we are referring to version 4 constructions.
Open a new sketch and construct a segment. Label the endpoints, say A and B. Do not worry about the length of the segment. Remember that we can drag it and make it either smaller or larger. The third tool (from top to bottom) is the COMPASS TOOL and is used to construct circles. Since every point of the circle is equidistant from the center, we can use circles to construct congruent segments. We will use two circles with the radius equal to the length of the segment you just constructed to determine the third vertex of the triangle. Select the COMPASS TOOL and move the mouse close to one endpoint. You will notice that the point is surrounded by a blue circle and the left bottom corner says "...from Point..." Click. Now, move the mouse to the other endpoint -- a circle should appear once you move the cursor -- and click once you notice it is selected. See Figure 8.
By now you should know what indicates the selection. Now construct
a circle reversing the order of the endpoints of the segments. The COMPASS
TOOL stays selected unless you click on a different tool. Since the third
vertex is the intersection point of the two circles, we need to select the circles
and then ask the software to construct the intersection point. To select
both circles, click on the SELECTION ARROW TOOL (the top tool) and click on
any region of the screen. Next click on both circles (A thicker line on
each circle indicates this selection. This quality of lines is only available
in version 4. Previous versions put small marks -- usually small squares
-- in the objects. Also, in version 3, you need to press and hold the
shift key if you want to select several objects). Now, click on the Construct
menu (fourth menu from left to right on top of the screen) and select Intersections
(see Figure 9).
Since the circles intersect in two places, the software constructs both points. Select one of the intersections and label it C. Now, construct segments AC and BC. Refer to Example 1, Constructing Segments, if necessary.
Now there is some cleaning up to do. However, there is a word of advice. In common language we might say that we "erase" the auxiliary constructions (in this case the circles and the other intersection points). This is not a good idea, since the word "erasing" can be confused with "deleting." If we delete the circles, the intersection points are also deleted and there is no triangle. What we have to do is "hide" the auxiliary constructions. To do so click on the SELECTION ARROW TOOL and then click on the two circles and the unused intersection point. Notice the comments on the bottom left corner of the screen. Once you select the three objects, the bottom right corner indicates so. Next, click on the Display menu (third from left to right on top of the screen, figure 10) and select Hide Objects.
Figure 10. Display menu.
Aha! An equilateral triangle remains.
We use the dragging properties of the software to test that this is an equilateral triangle. Since the SELECTION ARROW TOOL is still selected, click on vertex A (or B) and without releasing the mouse drag the vertex. As you notice, all three segments change but remain congruent since that is the geometric property that we used. A common mistake users make is not to use geometric properties while making constructions. Instead users use visual clues. For instance, they try to make an equilateral triangle by drawing three segments that "appear" to be congruent. But when users drag one of the vertices of the "equilateral" triangle they drew, they realize that the sides do not continue to be congruent. We refer to this situation as a construction that collapses. What really makes the triangle continue being equilateral is that we used its properties to construct it. In contrast the drawing (or sketching) does not use the properties. The drawing simply "makes the triangle to look equilateral." The use of the properties assures that the triangle is equilateral. You will find the distinction between a construction and a drawing very useful when introducing your students to any dynamic geometry software.
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