Right Triangle
When I first began to work on this problem I started with constructing a right triangle. I did this by first constructing segment c4 then constructing a line perpendicular to the segment, b4. From here I plotted a point on the perpendicular line and connected the end point of the original segment to the plotted point. This gave me a right triangle. Then I needed to construct right triangles off from the original triangle where the hypotenuse of the right triangle was the shared side. The first thing I did was to construct a line through point C2 that was parallel to segment c4. Next I constructed a line that was perpendicular to c4 and went through point B2. I constructed the intersection of these two lines at point N. and constructed segments b6, t1, and n1. I repeated this two other times for the other sides of the right triangle. Then I found the areas for the different triangles. I saw that there was a pattern. It was the same as the pervious patterns in that sum of the areas of the right triangles off of the small and long legs of the right triangles was the area of the right triangle off of the hypotenuse. Next I wanted to change the size of my right triangle to observe if this would hold true for a different size. When I tried to manipulate my right triangle I saw that the side lengths would increase and decrease, but the angle measures did not change. Also in class, Ms. Haltiwanger stated that this conjecture should not hold true for this problem. I realized that the construction I had was incorrect. I needed to build my right triangles so that their angles could change.
Next I tried to build my right triangles using circles. First I constructed a circle and found the radius and labeled is a. Next I plotted point A on the circle and connected the points so that I had a triangle with segments from B to A=c, A to C=b, and C to B=a to construct another triangle. Next I plotted point D on the circle and connected point C to D and D to B to construct another triangle. Then I constructed a circle with b as its radius. I constructed point E on the circle and constructed a triangle. I did the same and constructed a circle with a radius of c and plotted point F on the circle and constructed a triangle. Next I found the areas of each figure. When we move the right triangle in the center, the angles and side lengths change. After much thought and discussion with Kelsey, we could not formulate a relationship between the triangles. There only seemed to be a relationship in the first sketch that I created. After more thought I noticed a pattern. In the other parts of the problem, with the square and equilateral triangles, there was the same pattern. Where the area of the two smaller shapes summed to the area of the larger shape. The similarity in these problems is that all of the shapes built off of the right triangle have equal angles. The angles in all three equilateral triangles measure and the angles of the squares all measure . This is also true for my first sketch. All of the small angles measure , all the large angles measure , and they all contain a right angles. Also all of the sides are proportional to each other. With this information we can say that all of the equilateral triangles, squares, and right triangles are similar. After some more research I found the following link
http://jwilson.coe.uga.edu/EMAT6680Fa2012/Smith/6690/pythagorean%20theorem/KLS_Pythagorean_Theorem.html
It explains this idea further. So we can make a conjecture that when the shapes built off of a right triangle are similar, which means their angle measures are equal and their sides lengths are proportional, then the areas of the two figures built off of the legs of the right triangle sum to the area of the figure built off of the hypotenuse of the right triangle.