The bases of the trapezoid are equal to 3 and 11. Find the largest of the line segments into which

The bases of the trapezoid are equal to 3 and 11. Find the largest of the line segments into which one of its diagonals divides the middle line of this trapezoid.

Given a trapezoid ABCD; Base AD = 11; BC = 3;

Diagonal AC crosses the middle line EF at point N, you need to find NF;

Drop the perpendiculars BH and CM to the base AD; CM crosses EF at point O; HM = 3 since HBCM is a rectangle;

We denote AH = x, then AM = 3 + x: MD = 11 – 3 – x = 8 – x;

NO is the middle line of triangle CAM, so it is equal to half the base of AM; AM / 2 = (3 + x) / 2;

OF – the middle line of the CMD triangle, so it is equal to half the base of the MD; MD / 2 = (8 – x) / 2;

NF = NO + OF = (3 + x) / 2 + (8 – x) / 2 = 1.5 + x / 2 + 4 – x / 2 =  5.5.