Points A, B, and C are on a unit circle with A and C as end points of a diameter. The midpoint of chord AB is M. If the lengths of chord AB and line segment MC are the same, what is that length?

Draw segment BC. Angle ABC is a right triangle because AC is a diameter. Let AB=MC=x. Then AM=MB=x/2. Let diameter AC=1. Let BC=y.

Then by Pythagorean Theorem:

(x/2)^2 + y^2 = x^2 from triangle MBC

x^2 + y^2 = 1^2 from triangle ABC

The first equation can be rewritten as y^2 = (3/4)x^2. Substituting that into the second equation gives (7/4)x^2 = 1, or x = 2/sqrt(7) = 0.75593