I/D#2: Unit O- How Can We Derive the Patterns for a Special Right Triangle?

INQUIRY ACTIVITY SUMMARY
1. 30-60-90 Triangle
This equilateral triangle has all sides equalling one (because given) and the angles equalling 60 degrees. We know that because all angles in a triangle sum up to 180, and since there are 3 angles you divide 180 by 3 giving you 60 for each angle. However, to get our 30-60-90 triangle, we must cut the triangle vertically in half, having the top 60 degrees become two 30 degree angles and the middle bottom of the triangle two 90 degree angles as seen below. As for the sides equalling one, since we cut the triangle in half, each triangle is going to have the bottom side equalling 1/2. To find the other sides, we use the Pythagorean theorem which is a^2+b^2=c^2. In this case were solving 1/2^2+b^2=c^2 which results in b= radical 3 over 2. To make it easier and not deal with fractions we multiply everything by 2 and add a variable, n, to represent any number, meaning the relationship is consistent. We then get a=2n , b=radical 3 over 3, and c= n.

2. 45-45-90 Triangle
Since we want a 45-45-90 triangle, we cut the square diagnolly in half. How do we know we got two 45 degree angles? Well, a square has all 90 degree angles and when we cut it diagonally in half, the two 90 degree angles split into two, meaning the 90 degree angle is divided by 2 as seen below. We are given that all sides equal one but since we don't not know one of the sides because we cut the square we use the Pythagorean theorem which is a^2+b^2=c^2. In this case were solving 1^2+1^2=c^2, giving us c=radical 2. We then add n to each side in which represents any number meaning the relationship is consistent.

INQUIRY ACTIVITY REFLECTION 
1. Something i never noticed about a special right triangle is that there will always be two sides that are the same for example a 45-45-90 triangle a and b are the same and in a 30-60-90 triangle.
 2. Being able to derive these patterns myself aids in my learning because it gives me a better understanding of how everything connects. If i every forget the rules i can simply derive them.