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.