This has a pretty tough algebraic solution, but see the shortcut at the bottom:
m = 2(L + W), "what is m?"
Statement 1) LW = 60, so L = 60/W and W = 60/L
but it could be that (L,W) is (6,10) and 2(L + W) = 2(16) = 32
or it could be that (L,W) is (5,12) and 2(L + W) = 2(17) = 34
or even that (L,W) is (100000,0.0006) and 2(L + W) = 2(100000.0006) > 200000
Insufficient.
Statement 2) through the pythagorean theorem, L^2 + W^2 = 13^2 = 169, but it could be that (L,W) is (0,13) and 2(L + W) is 2(13) = 26, or that (L,W) is (12,5) and 2(L + W) is 2(17) = 34.
Insufficient.
Combined) we know that L = 60/W and that L^2 + W^2 = 169, so substituting we get (60/W)^2 + W^2 = 169.
So multiplying both sides by W^2 we have (60^2) + W^4 = 169W^2 and W^4 - 169W^2 + 60^2 = 0 or x^2 - 169x + 60^2 = 0 where x = W^2.
We can factor this as (W^2 - 144)(W^2 - 25) = 0, so W^2 = 144 or W^2 = 25 and W = {-12,-5,5,12}. The positive values that make sense in this problem are W = 5 and W = 12.
With a W of 5 or 12, the perimeter would remain the same at 2[W + (60/W)] = 2[5 + (60/5)] = 34 = 2[12 + (60/12)] = 34
Thus the statements combined are sufficient. The hard part is breaking down the quadratic that results from combining the statements from W^4 - 169W^2 + 60^2 to (W^2 - 144)(W^2 - 25) = 0.
You could also solve this with a shortcut:
m^2 = [2(L + W)]^2 = 4(L^2 + W^2 + 2LW).
With statement 1 we are given that LW = 60, so 2LW = 120.
With statement 2 we are given that L^2 + W^2 = 169.
So m^2 = 4(169 + 120) = 4(289) = (2^2)(17^2) = (34)^2, and m = {-34,34}, so the perimeter can be positive 34.
