Question 1197389
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There is no fixed length for either of these triangles.  Rather, there is a RELATIONSHIP between the hypotenuse and shorter two sides.  For example, for a 45-45-90 triangle, the two shorter legs have equal length, and we can call this length L.  If we do this, the hypotenuse h has length  {{{ h = L * sqrt(2) }}}.

This follows from the Pythagorean Theorem  where for any right triangle, the hypotenuse is related to the other two sides, a and b, by:  
{{{ h = sqrt(a^2 + b^2) }}}


Back to the 45-45-90 triangle:  
 {{{ h = sqrt(L^2 + L^2) }}}
 {{{ h = sqrt(2L^2)}}} 
 {{{ h = L * sqrt(2) }}}

So, if I pick L=10cm, h will be about 14.14cm
I can also pick L=23m, and h will then be 32.53m,  etc.
The important thing to note is both of these examples give you a 45-45-90 triangle.

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For the 30-60-90 triangle,  h is as before (square root of sum of squares of shorter two sides) but the relation of h to the shorter two sides is as follows:

   a = h * sin(30) = {{{(1/2)*h}}}
   b = h * cos(30) = {{{(sqrt(3)/2)*h}}}

where  a is the side opposite the 30 degree angle, and b is the side adjacent to the 30 degree angle (such that the 30 degree angle comes between 'b' and 'h').

Notice for the 30-60-90 triangle, the Pythagorean Theorem still holds, as it does for all right triangles:
      h = {{{ sqrt(((1/2)h)^2 + ((sqrt(3)/2)h)^2) }}}  
        = {{{ sqrt((1/4)h^2 + (3/4)h^2) }}}
        = {{{ sqrt(h^2) }}}
        = {{{ h }}}    


Hope this helps!