The numbers... they look familiar.

Plot on a graph the following points: (0, C), (0, C+A), (B, C). Join these points to get a right angle triangle with a hypotenuse of $\sqrt { { a }^{ 2 }+{ b }^{ 2 } }$

Then plot (B, 0), (B+D, 0), (B, C) and join them to get a right angle triangle with hypotenuse of $\sqrt { { c }^{ 2 }+{ d }^{ 2 } }$ .

We want to find the minimum distance of $\sqrt { { c }^{ 2 }+{ d }^{ 2 } } +\sqrt { { a }^{ 2 }+{ b }^{ 2 } }$ i.e. the distance between (B+D,0) and (0, C+A)

As given, b + d = 12, a + c = 5. The minimum distance between (0, 5) and (12, 0) is a straight line of length $\boxed{13}$

abcd

Let z1=a+i b; z2=c+i d;

Then, z1+z2=(a+c)+i*(c+d) =5+12i

Now, mod(z1+z2) is less than or equal to (modz1+ modz2)

So, |z1+z2|<=|z1|+|z2|

therefore, (5^2+12^2)^0.5<=(a^2+b^2)^0.5+(c^2+d^2)^0.5

and so, (a^2+b^2)^0.5+(c^2+d^2)^0.5>= 13 which is the answer

Can use the derivative logic to determine the numbers for minimum values ...........so have a=c and b=d ............you get twicw the root of 42.25 turns out to be twice 6.5 that is - 13( The lucky one this time :P)