Question

the distance between the center of symmetry of a paralleglogram and its longer side

Answer 1

Consider the parallelogram ABCD. Assume that we know its longest diagonal $d_1$and the distance between A and K, which is b according to the figure. Therefore, we can use the Pythagorean theorem.

Consider the right triangle based on the point O where the diagonals cross, points A and K. Its hypotenuse is half of the longest diagonal, catheti are the unknown distance and AK:

$$(d_1/2)^2=x^2+AK^2,\\ d_1^2/4=x^2+AK^2,\\ d^2_1/4-AK^2=x^2, \\x=\sqrt{d^2_1/4-AK^2}.$$