Equilateral Triangles
The above sketch is a right triangle with an equilateral triangle built off of each of the sides of the right triangle. In my sketch I also have the areas of each triangle shown. From these sketches I started to look and see if there was a correlation between the areas of the different triangles. First I looked at the relationships between the smallest equilateral triangle and the right triangle there did not seem to be any particular relationship between the sides. I recorded the different areas of the three triangles and then manipulated the size and recorded the values in the table below. I did this a few times to help to see the relationship between the areas of the triangles. From the table I can start to see a pattern. We can see that areas of the two smallest equilateral triangles add to the area of the largest equilateral triangle. In the first set of numbers we see that and the area of the third equilateral triangle is . We can also see that this is true for the second and third sets of triangles. We have and . From this we can make the conjecture that the ares of the equilateral triangles off of the legs of a right triangle add to the area of the equilateral triangle off of the hypotenuse of the right triangle. Lets test this theory. We can drag the right triangle so that it has an area of 15.19. From our conjecture the sum of the area of the two smaller equilateral triangles is equal to the are of the largest equilateral triangle. The areas of the two smaller equilateral triangles are 10.67 and 16.22. If our conjecture if correct then the area of the largest equilateral triangle should be . We can see that the area of the largest equilateral triangle is 26.89, and . Therefore our conjecture was correct.| Right Triangle | Equilateral 1 | Equilateral 2 | Equilateral 3 |
| 9 | 9.84 | 6.17 | 16.01 |
| 25.24 | 27.07 | 17.65 | 44.73 |
| 5.4 | 4.93 | 4.44 | 9.37 |