Simply consider the squares that lie on a main diagonal, each has a domino that goes through it, and each of these dominoes takes exactly 1 square from the main diagonal, in addition if you draw all possible centers consider the two biggest diagonals from this set of dots (that are parallel to our main diagonal chosen from the board), then the center of the unique domino in this fixed tiling for each square in the diagonal, lies in one of these two diagonals always, since there is $20$ squares in the diagonal we have $20$ centers that lie in both of these lines so one must have at least $10$ centers on it, thus we are done .