Curious Chord in a Circle

Geometry Level 5

\(A\) and \(B\) are points on the circumference of circle \(\Gamma\). \(M\) is the midpoint of the minor arc \( \stackrel{\frown}{AB}\) and \(C\) is a point on the circumference of \(\Gamma\) such that it is between \(M\) and \(B\). \(D\) is a point on the line segment \(AC\) such that \(MD\) is perpendicular to \(AC\). If \(AD = 45\) and \(BC = 15\), what is the value of \(CD\)?

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