Question 1201705
<pre>
We are not told which way the 30-60-90 triangle is turned.  So there are two
solutions.

A 45-45-90 triangle is half of a square cut down its diagonal.
A 30-60-90 triangle is half of an equilateral triangle cut through a vertex
perpendicular to the opposite side.

The properties of all 45-45-90 triangles are:
1. the two legs are equal 
2. the hypotenuse is {{{sqrt(2)}}} times the length of a leg.

The properties of all 30-60-90 triangles are:
1. the shorter leg is half of the hypotenuse. 
2. the longer leg is {{{sqrt(3)}}} times the shorter leg.

First solution (below):
Since the hypotenuse BC of triangle BCD is 24, its shorter leg, CD, is half of
24, or 12.
Since the shorter leg CD is 12, the longer leg BD is {{{12sqrt(3)}}}.

Since leg CD of triangle ACD is 12, the other leg AD is also 12, and the
hypotenuse AC is {{{12sqrt(2)}}}.

So {{{AC=12sqrt(2)}}} and AB = {{{12+12sqrt(3)}}}

{{{drawing(400,8800/43,-5,38,-5,17,
locate(1.8,2.7,45^o),locate(9.2,10,45^o),
locate(26.5,2.5,30^o),locate(12.5,10.5,60^o),locate(0,0,A), locate(6,0,12),
locate(19,0,12sqrt(3)),locate(22,8,24), locate(3,8,12sqrt(2)),locate(12.5,6,12),
locate(32.8,0,B),locate(12,0,D),locate(11.9,13.6,C),

triangle(0,0,12,0,12,12),triangle(12,12,12,0,32.8,0)      )}}}

Second solution (drawn below):
Since the hypotenuse BC of triangle BCD is 24, its shorter leg, BD, is half of
24, or 12.
Since the shorter leg BD is 12, the longer leg  CD is {{{12sqrt(3)}}}.

Since leg CD of triangle ACD is {{{12sqrt(3)}}}, the other leg AD is also
{{{12sqrt(3)}}}, and the hypotenuse AC is {{{12sqrt(3)}}} times {{{sqrt(2),
which gives {{{12sqrt(6)}}}.

So {{{AC=12sqrt(6)}}} and AB = {{{12+12sqrt(3)}}}

{{{drawing(400,12400/43,-5,38,-5,26,
locate(1.8,2.7,45^o),locate(18,18.8,45^o),
locate(28.5,2.7,60^o),locate(21.1,16.6,30^o),locate(0,0,A),
locate(10,0,12sqrt(3)),locate(27.6,11,24), locate(6.3,11.4,12sqrt(6)),
locate(32.8,0,B),locate(20.8,0,D),locate(20.6,22.4,C),
locate(21.2,10.5,12sqrt(3)),    locate(26,0,12),
triangle(0,0,20.8,0,20.8,20.8),triangle(20.8,0,20.8,20.8,32.8,0)      )}}}

Edwin</pre>