In a rectangular trapezoid ABCD, side AB is perpendicular to the bases AD and BC. Point K is the middle of the CD

In a rectangular trapezoid ABCD, side AB is perpendicular to the bases AD and BC. Point K is the middle of the CD side and AK is the bisector of the CAD angle. Find the ratio AD: BC if AB = 4 and AK = 6.

Since the bisector AK divides the CD side equally, ACD is an isosceles triangle and AK is the height. Lower the height CH perpendicular to the base of AD. CH = AB = 4. Right-angled triangles ADK and CDH are similar. KD / HD = 3/2, KD = 3 / 2HD. CD = 2KD = 3HD. HD² + 16 = 9HD². HD = √2. CD = 3√2. AD / CD = 3/2, AD = 9 / √2. In a triangle CAB AC = AD = 9 / √2, BC² + 16 = 81/2, BC = 7 / √2. So AD / BC = 9/7.