In square ABCD, AD is 4 centimeters, and M is the midpoint of CD. Let O be the intersection of AC and BM. What is the ratio of OC to OA? Express your answer as a common fraction.

Angle AOB = Angle MOC

And angle CMB = angle ABM

So...by AA congruency....

Triangle BOA is similar to triangle MOC

And since AB = AD = 4 and MC = 2 .....then the scale factor of triangle MOC to triangle BOA is 2 : 4 = 1 : 2

So

OC : OA = 1 : 2 = 1 / 2