Example 7. The Pythagorean Theorem (with an introduction to measuring areas).
The Pythagorean Theorem is one of the most important and beautiful results
in geometry. You may recall that it states that in a right triangle the
sum of the squares of the (lengths of the) legs is equal to the square of the
(length of the) hypotenuse. In Figure 27, 
, where a and b
are the legs and c is the hypotenuse of a right triangle. The Greeks
knew this theorem more than 2500 years ago and viewed it in terms of areas,
since 
, for instance, can be
interpreted as the area of a square whose side (length) is a. We
will use the software to illustrate this result geometrically. Open a
new sketch and construct a right triangle (see Example 4). Click
on the CUSTOM TOOL to access the Square script (If the sketch for Example
6 is still open, then the Square script is still available). Use
it to construct an outward square on each side of the triangle. The final
diagram should look like the one shown in Figure 27.
Figure 27
Ignore the labels and the orientation. Just make sure that the squares are correctly placed. Run the drag test before continuing.
Now we will measure the areas of the squares. One may think that selecting the sides or the vertices of each square, and then using the Calculate menu would suffice to obtain the area. However, this is not the case. We see and recognize a square, but the software does not. The software needs to know which enclosed area you want to use either for calculations or for another construction. So, choose any square, say the one on the hypotenuse. Select all four vertices (you do not need the segments, since the software understands that if you give two points then you are thinking as well of the segment joining them) counterclockwise (or clockwise) making sure that this is respected. Open the Construct menu and select Quadrilateral Interior (in previous versions of GSP you will see Polygon Interior). See Figure 28 to check how the square is filled in.
Figure 28
If the interior of the square does not look completely filled in, it is very likely that the order of the vertices was not respected. In this case, undo the previous step and reselect the vertices taking care to select them in a clockwise or counterclockwise direction. Next, open the Measure menu (Figure 28) and select Area. Notice that the software labels the vertices if not labeled and displays the area with labels and units. (Previous versions will not label the vertices, but will show the area with labels and units). Compute the area of the other two squares. Make sure that when you finish computing one area, you click outside the measurement (in this case, each area). Otherwise it remains selected and when you select the vertices of the next square, you will not be able to construct the quadrilateral interior. After constructing the interiors of the three squares and measuring their areas, you should end up with a screen like the one in Figure 29.
Figure 29
Although we could use a calculator to make computations, this is not practical since every time that we drag the vertices of the triangle, the squares, and hence the areas, change. Instead, we will use the built-in calculator in the software to make operations at the same time that the areas change. Without selecting any object, open the Measure menu and select the Calculate command. Observe that a screen that looks as a calculator appears (Figure 30).
Figure 30
We need to select the areas needed in the computation. In this case, we need to add the areas of the squares on the legs. However, the calculator device is in our way and it is hard to remember what labels correspond to each square. So click on any part of the blue region on top of the calculator, and while holding down the mouse button, move (literally!) the calculator screen until you can see the labels of the square. See Figure 31.
Figure 31
Now, click on measurement of the area of the square on one leg. Next, click on the "+" sign on the calculator. After this, click on the measurement of the area of the square on the other leg. If necessary, move the calculator screen until you see the OK command on the calculator (next to Cancel). This OK command functions as an "equal" or "Enter" command in calculators. Once you hit it, you will see a computation indicating what was computed (See Figure 32).
Figure 32
Even better, drag any vertex of the right triangle and you will appreciate the power of this dynamic geometry software. Even when the construction is moved around, stretched or shrunk, all properties used to construct it are preserved. This is a wonderful way to see several instances of the Pythagorean Theorem.