There are the only 2 angle based special triangles. 45 45 90 is special because it is isosceles as well as right. Using the Pythagorean theorem and the fact that it has two equal sides, we can find out the ratio of all sides.
Example: If the legs are length n, then the hypotenuse will be
sqrt(n^2 + n^2) <------ by the Pythagorean theorem. sqrt(2n^2) n * sqrt(2)
30 60 90 is special because it is a equilateral triangle cut in half. Therefore one of its sides is half the length of another (the hypotenuse) The third side can be found with the Pythagorean theorem.
An equilateral triangle has a length 2n. If we draw an altitude, it will bisect the triangle and create 2 congruent right triangles. The bases of the right triangles are n. The altitude is
sqrt((2n)^2 - n^2)
sqrt(4n^2 - n^2)
sqrt(3n^2)
n * sqrt(3)
It is also possible to find more "special" angle triangles whose angles are 15 75 90 with the angle addition formulas.
There are plenty more special right triangles whose nature is based on sides rather than angles, such as the 3-4-5 and 5-12-13. There is an infinite number of such triangles and they can be found easily.
It has already been proven that the difference between squares of numbers n^2 and (n+1)^2 is an odd number 2n+1. Because of this, all odd numbers are the difference of some pair of squares n^2 and (n+1)^2.
Choose any odd number and use it as the side of a right triangle. Because it is odd, its square is also odd. Because its square is odd, its square is the difference between a pair of squares. So if the remaining sides are n and n+1, then our chosen side will be sqrt(2n+1). Square our chosen odd number, subtract 1 from it and divide by 2 to get n, which is the length of the other leg. The final side is the hypotenuse and is n+1.
For example, let's try 21.
21 = sqrt(2n+1)
441 = 2n+1
220 = n
221 = n+1
21, 220, and 221 form a Pythagorean triple.
Check:
21^2 + 220^2 = 221^2
441 + 48400 = 48841 <----- TRUE!
