Since we divided the side which had previously equaled 1 into two equal parts, then each triangle must have a side of 1/2. So, if we proceed with that deduction, then that side, squared, added with the other side that's not the hypotenuse, squared, should equal the hypotenuse, squared. Basically, a² + b² = c² (Pythagorean Theorem), with 1/2 being a, the other leg being b, and 1 being c. This works because the triangles we had created, have 90° angles, and therefore, have an assumed hypotenuse across from them, along with the values for a and b that fit into the aforementioned equation. (1/2)² + b² = (1)² would be 1/4 + b² = 1, and subtracting 1/4 from both sides would result in 3/4 as the value for b². And, if we take the square root of the entire fraction, then it should equal √3/√4, which would reduce into √3/2.
Lastly, we'd have to save this extremely rare knowledge, by multiplying everything by 2, and then, by a variable, so that this pattern will be easily transferable to any 30/60/90 degree triangle. Multiplying each side by 2n, basically, would yield a hypotenuse of 2n, a value of n for the side across from the 30° angle, and a value of n√3 for the side across from the 60° angle. This pattern shall forever exist for any 30/60/90 degree triangle, since the pattern was derived from ratios that will be the same for all 30/60/90 degree triangles.
2) We may derive a 45/45/90 degree triangle's special pattern from a square with the equal sides of 1, by diagonally dividing two 90° angles (across from each other) exactly in half, and making four 45° angles - two for each triangle. Each triangle would then, each have two sides that equal 1, and share a hypotenuse.
So, since the two triangles that we have given life to have 90° angles, then the Pythagorean Theorem can be utilized in order to find the length of the hypotenuse in this set-up. 1² + 1² = c² shall become c² = 2, and from that point onward, be √2.
In order to again, preserve our newly-discovered and revolutionary (in the field of mathematics) pattern, we must multiply all of our triangles' values by n, in order to create a distinct pattern that we can follow for any 45/45/90° triangle. Everything just basically gets an n attached to it.
Inquiry Activity Reflection
1) Something I never noticed before about special right triangles is that their established patterns can be deduced from a logical procedure, instead of just being "there", because we were told that it happened to be "there".
2) Being able to derive these patterns myself aids in my learning because I'm not just remembering given facts or something - I'm actually finding out this stuff myself.
