sorting - SortBy misbehaving?

It maybe a stupid thing in the end but I'm stuck a couple of hours now.

I have a list of 2 points on the plane and I want to get the one with the biggest second coordinate. I thought I knew how SortBy operates. For example

SortBy[{{a, 1/2 (2 + Sqrt[2])}, {b, 1/2 (3 + Sqrt[2])}, {c, 1/2 (1 + Sqrt[2])}}, Function[{x}, x[[2]]]]

gives as expected the answer

{{c, 1/2 (1 + Sqrt[2])}, {a, 1/2 (2 + Sqrt[2])}, {b, 1/2 (3 + Sqrt[2])}}

and

SortBy[{{a, 1/2 (2 + Sqrt[2])}, {b, 1/2 (3 + Sqrt[2])}, {c, 1/2 (1 + Sqrt[2])}}, Function[{x}, -x[[2]]]]

gives as an answer

{{b, 1/2 (3 + Sqrt[2])}, {a, 1/2 (2 + Sqrt[2])}, {c, 1/2 (1 + Sqrt[2])}}

which is perfectly fine.

My list is

{{4/13 (-9 - Sqrt[3]), 6/13 (4 - Sqrt[3])}, {4/13 (-9 + Sqrt[3]), 6/13 (4 + Sqrt[3])}}

and the command

SortBy[{{4/13 (-9 - Sqrt[3]), 6/13 (4 - Sqrt[3])}, {4/13 (-9 + Sqrt[3]), 6/13 (4 + Sqrt[3])}}, Function[{x}, x[[2]]]]

gives the answer

{{4/13 (-9 - Sqrt[3]), 6/13 (4 - Sqrt[3])}, {4/13 (-9 + Sqrt[3]), 6/13 (4 + Sqrt[3])}}

which is the same as the answer I get from

SortBy[{{4/13 (-9 - Sqrt[3]), 6/13 (4 - Sqrt[3])}, {4/13 (-9 + Sqrt[3]), 6/13 (4 + Sqrt[3])}}, Function[{x}, -x[[2]]]]

Can someone explain to me what's going on? This drives me crazy.

Answer

I don't know exactly why Mathematica is giving a wrong result, but here's a workaround:

SortBy[{{4/13 (-9 - Sqrt[3]), 6/13 (4 - Sqrt[3])}, {4/13 (-9 + Sqrt[3]), 6/13 (4 + Sqrt[3])}}, 
        -N @ #[[2]] &]

That is, force Mathematica to sort by their numerical value.

OR you can use Sort instead:

Sort[{{4/13 (-9 - Sqrt[3]), 6/13 (4 - Sqrt[3])}, {4/13 (-9 + Sqrt[3]),
    6/13 (4 + Sqrt[3])}}, #1[[2]] > #2[[2]] &]

Which gives the same result.