What are the basic properties of a 45-45-90 triangle?

The basic properties are: An interior angle of ##45^@##, an exterior angle of ##135^@##, an opposite angleof ##45^@##, a right angle (##90^@##), opposite side length: ##sqrt2/2##, and adjacent side length of ##sqrt2/2##, and a hypotenuse of 1. (side lengths for a standard triangle inside of the Unit Circle)

Explanation

All standard right triangles inside of the unit circle have a hypotenouse of 1. This is because the unit circle is defined as a circle with a radius of 1. Thus we create Unit Circle Triangles by picking a radius, and drawing its vector components.

In this case, we know we have a triangle with a primary angle of ##45^@## and, as mentioned before, a hypotneouse of 1. As such we can use ##sintheta## and ##costheta## to determine the other two lengths.

##sintheta=o/h->hsintheta=o->1*sin(45^@)=sqrt2/2##

##costheta=a/h->hcostheta=a->1*cos(45^@)=sqrt2/2##