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?
- Create
a custom tool for the construction of a right triangle if you have not already done so.
Then use your custom tool to construct a right triangle in your sketch
window. (Instructor
Note: A
Sketchpad file illustrating the construction of a right triangle has been saved as
righttri.gsp.)
- 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
from the toolbox.
- Position the hand
pointer on any label.
- Double click and
type the new label in the Label 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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- Manipulate
the triangle
around your sketch window by dragging one of its vertices. Observe changes in
the lengths of sides and area of the triangle. Use these measurements to verify the formula for the
area of a right triangle.
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 custom tool for the construction of a square.
In a New Sketch window,
create a custom tool for the construction of a square.
Hint:
Combine your techniques from constructing an isosceles triangle
and a right triangle. (Instructor
Note: A
Sketchpad file illustrating the construction of a square has been saved as
square.gsp.)
-
Manipulate your square.
How do each of the vertices move when you drag them around your sketch
window? Which vertices
are the most restricted and which are the least restricted? As you manipulate
your figure, does it always stay a square? How does your construction
guarantee that your square will remain a square?
Use your
custom tool to create several squares in your sketch
window. Pay close attention to
how the custom tool creates a square.
Now we are ready to construct squares on the sides of a right
triangle. In a New
Sketch window and using your custom tool, 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 vertices in the wrong order. Under the Edit menu, select Undo Square. Try
using your custom tool again, keeping in mind that the
order of your vertices 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?
(Instructor
Note: A
Sketchpad
file illustrating the
Pythagorean
Theorem
has been saved as
pythag.gsp.)
To
construct the polygon interior of a quadrilateral:
- Select the vertices
of the quadrilateral, in order.
- Choose from the Construct
menu, the Quadrilateral Interior command.
- Click anywhere inside
the sketch window to de-select the quadrilateral 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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- Formally state and prove the Pythagorean
theorem.
Part 3:
- Under the File
menu choose Open...
In the new window, look in the C:. Double-click on
Program Files
» Sketchpad
» Samples
»
Sketches »
Geometry » Pythagoras.gsp.
- Click
on the Behold Pythagoras! button in your sketch window. 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.
- Click
on the Contents button and then the Leonardo da Pythagoras
button in your sketch window. 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.
- Click
on the Contents button and then the Puzzled Pythagoras
button in your sketch window.
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.
- Click
on the Contents button and then the Shear Pythagoras!
button in your sketch window. 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.
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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