The key to this problem is recognizing that the line DF must bisect the angle BDC. This is because CD must coincide with diagonal BD when folded along line DF.
The angle BDC is clearly 45 degrees since the diagonal bisects the right angle.
Therefore angle x = 45/2 (Attached diagram).
You can then find CF (labeled y) using tangent function.
[tex]tan (\frac{45}{2}) = \frac{y}{1}[/tex]
From half -angle formula
[tex]y = tan (\frac{45}{2}) = \frac{-1 \pm \sqrt{1+tan^2 (45)}}{tan (45)} = -1 \pm \sqrt{2}[/tex]
Eliminating the negative case since y>0.
[tex]y = -1 + \sqrt{2}[/tex]
Therefore
k = 2
m = 1
Final Answer:
k+m = 3
