The bases of a trapezoid are 17 and 19. Find the largest of the line segments into which

The bases of a trapezoid are 17 and 19. Find the largest of the line segments into which one of its diagonals divides the middle line of this trapezoid

Let the trapezoid ABCD be given with the bases BC = 17 and AD = 19. Through the points M and N, which are the midpoints of the lateral sides of the trapezoid AB and CD, the middle line of this trapezoid MN is drawn. Let’s draw the diagonal of the trapezoid AC, which will intersect the midline at point K. By Thales’s theorem, we find that this point divides the diagonal AC into two equal segments AK = KC, then the midline segments MK and KN will be midlines in Δ ABC and Δ ACD, respectively , and MK <KN, since BC <AD. Let’s find the largest of the segments into which its diagonal divides the middle line of this trapezoid: КN = АD: 2; KN = 19: 2; KN = 9.5. Answer: the largest of the segments into which its diagonal divides the middle line of this trapezoid has a length of 9.5.