Part 1:
- State and illustrate the
Pythagorean theorem. Discuss and interpret the meaning of the algebraic
equation, a2 + b2 = c2. What do a,
b and c represent? What do a2, b2 and c2
represent?
- Script the construction
of a right triangle if you have not already done so.
(Instructor
Note: A
sketchpad script illustrating the construction of a right triangle has been saved as
righttri.gss.)
- Does the Pythagorean
theorem
hold true for your triangle? It will probably be easier for you to relabel
your triangle's sides and vertices to follow the usual conventions (i.e.,
vertices A, B, C where the sides opposite these vertices are a, b, c; and c is the hypotenuse). Using the Sketchpad measurement tools and
calculator, numerically verify the Pythagorean theorem.
To
relabel an already labeled figure:
- Select the text
tool (the hand) from the toolbox.
- Position the finger
on the label.
- Double click and
type the new label in the Relabel dialog box.
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- Drag a vertex (or side) to further verify the theorem.
- Measure the area of your right triangle.
To
construct the polygon interior
of a closed
figure:
- Select the vertices
of the closed figure, in order.
- Choose from the Construct
menu, the Polygon Interior command.
- Click anywhere inside
the sketch window to de-select the polygon interior.
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To measure the area of a polygon:
- Select the polygon interior.
- From the Measure menu,
select the Area command.
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Part 2:
-
Discuss how can you think about the Pythagorean
theorem in
terms of areas of squares.
-
To construct squares on each side of your right triangle, it
would be helpful to have a script for the construction of a square.
In a New Sketch window,
script the construction of a square.
Hint:
Combine your techniques from constructing an isosceles triangle
and a right triangle. (Instructor
Note: A
sketchpad script illustrating the construction of a square has been saved as
square.gss.)
We have created four different
scripts for the constructions of an equilateral triangle, an isosceles
triangle, a right triangle and a square.
There is an easy way to access all of these scripts without having
to open and close multiple windows each time you wish to use them.
- Open your four scripts
to add a descriptive comment to each. On a disk, re-save each script
in a new folder named "GSP Scripts".
To add a comment to a script:
- From the Edit menu, select Show Comment.
- In the script window, type a descriptive comment in the box.
- Save your script.
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- Set the Sketchpad's Script Tool directory to this new folder.
To set Sketchpad's script tool directory:
- Choose Preferences… from the Display menu.
- Click on the More… button at the bottom of the window.
- Click on the Set button.
- Locate and select the GSP Scripts folder you created above,
click OK.
- Click on the Continue button in the More Preferences
window.
- Click on the OK button in the Preferences window.
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Notice that a new tool appears
in the toolbox. You can put any number of scripts in the script tool directory;
however, you will need your disk each time you work with Sketchpad to
access your script tool directory. If you are working on your personal
computer, you might want to save your script tool directory to a folder
on your hard drive.
-
Playing back scripts using your script tool is different than playing
back scripts opened individually in a window. Click on your script
tool, and select the square script. Refer to the Tool Status
Box to identify the necessary "Givens". Click anywhere in the sketch
window to activate the script. Click again to conclude your script of
a square.
Use your script tool to create several squares in your sketch
window. What happens when
you choose your “given” points in reverse order?
Make sure you are comfortable with the necessary selection
order of the given points before you move on.
Now we are ready to construct squares on the sides of a right
triangle. In a New
Sketch window, construct a right triangle.
Further construct squares (that do not overlap your right
triangle) on each side of your right triangle.
If a square overlaps the triangle, don’t worry.
You have selected the two given points in the wrong order. Select the segments and vertices of the overlapping square
and choose Hide Objects
under the Display menu. Try playing back your script again, keeping in mind that the
order of your point selection is important.
Measure the area of each of the three squares.
What do you notice about the areas?
Manipulate your triangle.
What conclusions can you draw from your observations?
Formally state and prove the Pythagorean
theorem.
Part 3:
- Under the File
menu choose Open...
In the new window, direct your attention to the Directories
listed. Double-click on samples; then sketches;
then pythag.
- In
the Files column,
open shear.gsp from the
Pythag Directory. Shearing
is a transformation which translates every point in a figure in a
direction parallel to a given line by a distance proportional to a
point’s distance from the line.
In your sketch window, drag point P to shear the parallelogram
back and forth. Note that
the parallelogram’s area doesn’t change as you change the
figure’s shape. Finish
the shear by dragging P to lie on the red line.
Explain why shearing does not affect the area of the
parallelogram. Shear the
other parallelogram on the other leg of the triangle.
Drag until point Q is on the line.
Measure the area of each of the three squares, and then follow
the directions given. Comment
on your observations. Prove
that the two shaded figures are congruent.
- In the Files
column, open behold.gsp
from the Pythag Directory. The twelfth century Hindu scholar, Bhaskara
demonstrated the Pythagorean theorem with a similar figure to the one
found in your sketch window. The
only text accompanying the figure was the word “Behold!”.
Drag point D and observe what happens to the right triangles in
the square on the left. Do
the interior figures change shape?
How is the hypotenuse of one of the triangles related to the
side of the original square? Can
you describe the area of this square in terms of the hypotenuse?
Verify algebraically that the area of the five figures sum to
the area of the larger square. Hint:
To start, write an expression for (1) the area of the whole
square in terms of c; (2) the areas of the four right triangles in
terms of a and b; and (3) the area of the interior square.
You should be able to write an equation involving a, b, and c.
- In the Files
column, open leonardo.gsp
from the Pythag Directory. Leonardo da Vinci (1452-1519) was a great
Italian painter, engineer, and inventor during the Renaissance.
He was also an amateur mathematician credited with the
following proof of the Pythagorean theorem.
Before you click on any of the buttons, describe what is
displayed in the sketch window. What
do each of the regions of the figure represent?
Double click on the “reflect top” button.
What happened? Did
the area of the regions change? Double
click on the “turn D’s” button. Again,
describe what happened. Further
explore this sketch and interpret in your own words Leonardo’s proof
of the Pythagorean theorem.
- In the Files
column, open puzzled.gsp
from the Pythag Directory. Follow the directions within the sketch
window. Can you explain
how this demonstrates that the area of the square on the hypotenuse is
equal to the sum of the areas on the squares on the other two legs of
the triangle? This dissection demonstration convincingly illustrates the
truth of the Pythagorean theorem, but does not provide a formal proof
of the theorem. Using the
“Puzzled Pythagoras” sketch and an argument of congruent
triangles, formally prove the Pythagorean theorem.
Extensions:
-
State the converse of the Pythagorean theorem. How could
you investigate the converse of the Pythagorean theorem? One way is by
constructing squares on the sides of an arbitrary triangle. Measure the
areas of the three squares and calculate the sum of two of them. Then
drag a vertex until the sum equals the area of the third square. What
kind of triangle do you have? Try to formally prove the converse of the
Pythagorean theorem.
Construct a right triangle and place a non-overlapping
equilateral triangle on each of its sides. Conjecture a relationship
between the areas of the equilateral triangles.
Measure the areas to assess your conjecture. Formally prove your
conjecture. Try placing
other figures (e.g., pentagons, hexagons, semicircles, etc.) on the
sides of a right triangle. Conjecture
a relationship between the areas outside of the right triangle. Then
measure the areas to assess your conjecture.
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