Question 1177803
The legs of a right triangle are in the ratio 1:2.<pre>
That means the right triangle is similar to this right triangle

{{{drawing(200,150,-1,3,-1,2,
triangle(0,0,2,0,2,1), locate(1,0,2),locate(2.05,.5,1), locate(1,1,sqrt(5)))}}}

</pre>The angle bisector to the short leg divides it into two
segments, one of which is 1cm longer than the other.<pre>

We draw the angle bisector (in green), and let the two segments be 
x and x+1:

{{{drawing(400,224,-5,45,-5,23,

triangle(0,0,35.88854382,0,35.88854382,17.94427191),
green(line(0,0,35.88854382,8.472135955)),
locate(36.5,5,x),locate(36.5,14,x+1)
 )}}}

A famous theorem says:

The bisector of an angle of a triangle divides the opposite side into
segments that are proportional to the adjacent sides.

We also know the ratio is 2:√5 from the similar right triangle at the top.

{{{x/(x+1)=2/sqrt(5)}}}

Solve for x and get

{{{x=4+2sqrt(5)}}}

So the right side of the triangle is {{{(4+2sqrt(5))+(5+2sqrt(5))}}} = {{{9+4sqrt(5)}}}

Since the side corresponding to 1 is 9+4√5, we know the scale factor is
9+4√5.

Since this triangle (the one at the top)

{{{drawing(200,150,-1,3,-1,2,
triangle(0,0,2,0,2,1), locate(1,0,2),locate(2.05,.5,1), locate(1,1,sqrt(5)))}}}

has perimeter 3+√5, we just multiply that by the scale factor:

{{{(3+sqrt(5))(9+4sqrt(5))}}}

and get 

{{{47 + 21sqrt(5)}}}   <--answer

Edwin</pre>