As shown in the figure, MN is the middle vertical line of AA', so AD = A'd.

AD+BD=A'D+BD

When BDA is on the same straight line, AD+BD=A'D+BD is the smallest.

According to the geometric conditions, the minimum value is the root number 2?

Oh, I typed P as D by mistake. I'm just looking around, and it's troublesome to change the map again.