Tori Bennett
Mate 4001
31 October 2012
Artifact 2: Geometer’s Sketchpad
Statement of Mathematical Sketchpad:
For artifact 2 exploration I will be using Geometer’s Sketchpad, commonly known as
GSP, to investigate the capabilities to prove the Pythagorean Theorem by using
Leonardo da Vinci’s proof. By using squares, midpoints, and the tools in GSP I will be
able to construct a proof that demonstrates the proof developed by Leonardo da
Vinci. This proof which will be explored is as follows:
Let A1 be the sum of the areas of the two squares with sides of lengths a and b,
respectively, together with the two triangles that share sides with these squares.
Let A2 be the sum of the areas of the square with side lengths c, together with the
two triangles that share sides with that square. In both A1 and A2, the dashed
segment divides the figure into two congruent parts that have area ½(a+b)2-½(ab)=
½(a2+ab+b2). Then A1=A2. Also, A1=a2+ab+b2 and A2=c2+ab. Therefore a2+b2=c2.
Exploration:
To start the exploration I need to construct a right triangle with squares on the legs
and connect the corners of the squares to construct a second right triangle.
Constructing the right triangles begins by drawing any length segment on a new
document of sketchpad. To ensure that the triangle has a 90-degree angle you rotate
the segment constructed 90 degrees around point A. Then connect these two
segments to create the hypotenuse.
To construct the
two squares on the legs rotate segment BA around point A and point B by 90
degrees. Do the same for segment AC around point A and C.
Construct points on the ends of the new segments and connect them using the
segment tool (creating two squares).
segment that connects points D and E.
Construct a
With this picture
developed construct a segment through the center of the figure (connecting points G
and F). Make the segment dashed by selecting the segment then click display, line
style, dashed.
Next, I need to construct a midpoint on the
dashed segment. To do so, select the segment then click construct, midpoint.
The segment GF divides the figure into mirror
halves. Select all the segments and points on one side of the center and create a
‘hide/show’ action button. To create this button select segments and points then
click edit, action buttons, hid/show. A button will appear on your screen, click the
button and notice how the segments selected disappear and the button changes to
say ‘show objects’.
Next, I need to rotate the entire
figure around point A (the midpoint). To do so, highlight point A, click transform
then mark center. To rotate the figure select all the segments and points except the
dashed line.
I want to create an action button like
the one we created before. Select the segments created by the rotation and click edit,
action buttons, hide/show. Rename the button to say ‘hide rotation’ and ‘show
rotation’, to do this right click, properties, label, and rename to say ‘hide rotation’.
Hide the rotation.
Construct a segment that connects
point C’ to point B and point B’ to point C.
This creates c squared! Next to show that it creates a square we will construct the
interior of the polygon of C’B and B’C to construct the interior select all the points
involved (points C’, B’, B, and C). Click construct, then quadrilateral interior.
Next I want to create the interior of the
triangles on the end. Follow the steps as before to construct the interior (do the
triangles separately) select points G, B, and C’, construct, triangle interior. Select
points B’, C, and F, construct, triangle interior.
Select the segments C’B and B’C and the three interiors and create a ‘hide/show’
button (constructed before), click edit, action buttons, hide/show.
Investigate: Explain to a classmate or make a presentation to the class to explain
Leoardo’s proof of the Pythagorean Theorem.
Through this exploration I can conclude that the four quadrilaterals are equal.
Because the measure of angle B is 45 degrees. This is because C’GB is right-angled,
thus the center T of the square C’B’CB lies on the circle circumscribing triangle C’GB.
Now, area(GFBC’)+ area(FGBC)= area(GBCF) + area(A’B’C’G). Each sum contains two
areas of triangles equal to C’BG removing which one obtains the Pythagorean
Theorem.
Prove: Write a paragraph that explains why the two hexagons have equal areas and
how these equal hexagons prove the Pythagorean Theorem.
The side lengths of the hexagons are identical. The angles at C (right angle + angle
between F & B) are identical. The angles at B (right angle + angle between G & C) are
identical. Therefore all four quadrilaterals are identical, and, therefore, the hexagons
have the same area.
Conclusion:
By exploring using GSP I have been able to prove that Leonardo da Vinci’s proof
does indeed prove the Pythagorean Theorem. From this exploration we learned how
to develop polygon interiors, construct squares and triangles using rotation and
creating action buttons. Through the exploration I was able to better discover the
capabilities of GSP’s tools. Geometer’s Sketch Pad has been a useful aid in the
visualization aspect of the proof.
For many students proofs are discouraging; when they see the word ‘proof’ they
dread the rest of the lesson. By using GSP an instructor is able to make proofs more
fun and interactive for students. The proof has the capability of being more than just
words that students commonly do not follow. By figures and transformations GSP
helps the proof come to life.