Example 6. Constructing a square (with more about measuring and an introduction to the Script Tool).
Now we construct a square. Remember that a square is a quadrilateral
(four-sided polygon) with four congruent sides and four congruent angles.
Open a new sketch and construct a segment AB. Since the sum of the interior
angles in a quadrilateral is 
, each angle must be a
right angle (i.e., its measure is 
). Therefore, the
square has perpendicular congruent sides. Because we usually label geometric
constructions counterclockwise, it is important that you follow the directions
given next in the same order as they are presented. Also, label the objects
you need after you hide the auxiliary constructions. In general, this
is very helpful since it avoids selecting an unintended construction.
Construct a segment BC congruent and perpendicular to segment AB through B, and a segment AD congruent and perpendicular to segment AB through A (consult Example 5). In our particular construction, do not label vertices C or D before you hide auxiliary constructions in the following directions. Join vertices CD. See Figure 17.
Figure 17
Hide any auxiliary constructions so that there is only square ABCD left. Label the vertices. Use the drag test to check that you have a square. You could also measure (consult Example 5) the length of the sides of the square to check that they are congruent. So even if the sides "do not look nice," make sure that they are congruent (by having the same length). You can also measure each angle and check that they are congruent. In this case, you need to select three vertices. To make this easier, select vertices A, B, C (in that order), open the Measure menu and click on Angle. See Figure 18.
Figure 18.
Then do the same for angles 
, 
and 
. As you will notice
the vertex in the middle is the common endpoint of the two rays that determine
the angle. For instance, if you use vertices ABC this means that you are
measuring the angle determined by the rays BA and BC. After you measure
the angles, run a drag test to make sure that the angles remain congruent.
It is worth it to mention that the software only
provides measures of angles from 
to 
.
One of the extraordinary capabilities of GSP is the ability to record the steps to make a construction. This recording, called a Script and saved using the SCRIPT TOOL, allows one to repeat the same construction over and over.
Select the four vertices and the four segments of the square. Open the CUSTOM TOOL (it is the bottom tool on the left). See what it looks like in Figure 19.
Figure 19
Click on Create New Tool. A similar screen to the one in Figure 20 should appear.
Figure 20
In the Tool Name box, type Square to replace whatever is on it (in this Case Tool#1). Click on OK.
The CUSTOM TOOL contains special features that may positively affect both your learning and teaching of mathematics. Figure 21 shows what is inside of this tool: Tool Options and Show Script View on top, and the tools created, in this case Square, on the bottom.
Figure 21
The Tool Options permits one to view the tools created and to delete the unwanted ones. You should open it to see what it looks like. The Show Script View allows one to see every step used in the construction, in this case, of a square. Click on it and you will see a screen similar to the one shown in Figure 22.
Figure 22
On the top of this window, you can write your comments. Next, there are the givens: In this case two points A and B. Remember that we started out with a segment AB. The software "knows" that you need to give it two points since they will determine the segment AB. The rest, congruent and perpendicular sides will depend on these givens. Next, the steps used in the construction appear. Scroll down so you can see them. On a related pedagogical point you can maximize the script, print it out, and give it to the students. Your students then have every step you want them to perform.
At the very bottom appears what is needed to run the script. In this case, it says, "To apply to selections, first select a point to match Given Point A". Close the Script (click on the right top box). Click on the CUSTOM TOOL. Since we have only one script, we do not need to indicate which script we will use. Once you click on this tool, you will notice that the bottom left corner of the sketch says "1. Match point A." If you move the mouse on the screen, you will see a red dot. This means that the software has given you the option of constructing a point. You can click on any part of the screen. If you move the mouse you will see that the software not only creates a segment, but, at the same time, it creates a square! See Figure 23.
Figure 23
This square appears selected and the Square script continues to be selected. In case that you do not like the square where it is, you can use the Undo command in the Edit menu to delete it completely.
One needs to remember the order in which the vertices A and B are selected. Suppose that we want to construct an outward square ABCD on side BC. If we select vertex B first and then try to select vertex C, the new square will overlap with the original square (see Figure 24)
Figure 24
On the other hand, if we select vertex C first and then vertex B, we obtain the desired square (see Figure 25).
Figure 25
Use the Square script to make a diagram like the one shown in Figure 26.
Figure 26
Ignore the labels. Once you have finished, click on the SELECTION ARROW TOOL and drag either vertex A or B. How about that? Keep this sketch open since we will use it in the next construction.