I/D#2 unit O-How can we derive the patterns for our special right triangles?

For the 30-60-30 triangle label all the sides by 1. Divide the triangle in two. That will divide the 60 degrees in to 30 degrees, the adjacent in half and make a 90 degree. Use the Pythagorean theorem to get the opposite side. Multiply by 2 to leave the radical alone and get 1 in the adjacent side and get 2 in the hypotenuse side. 

 For a 45-45-90 triangle label all the sides by 1. Divide the square to get two triangles. That will divide the 90 degrees into 45 degrees. Use the the Pythagorean theorem to get the hypotenuse side.  

 We use "n" to reference the missing value.

Something I never noticed before about special right triangles is, why we use "n" for?

Being able to derive these patterns myself aids in my learning because it gives a picture of both triangles and its formula. 

This are pictures of both triangles and showing how to get their formulas.

30-60-30 triangle

45-45-90 triangle