Chords AB and CD are perpendicular to each other and intersect at point M, the distance between the midpoints

Chords AB and CD are perpendicular to each other and intersect at point M, the distance between the midpoints of these chords is equal to a. What is the distance from point M to the center of the circle.

Let the points H and K be the midpoints of the chords AB and SD. Let us draw from the center of the circle O the perpendiculars OH and OK to the chords AB and CD.

By condition, the chords AB and BC are perpendicular, the angle CMD = 90 by condition, and the angles OHM and OKM are 90 by construction, then the quadrilateral OHMK is a rectangle, and the segments OM and HC are its diagonals.

Since the diagonals in the rectangle are equal, then OM = HK, as required.