There are a few relations related to tangents and secants that you are expected to remember. These problems make use of those.
1. When chords cut each other, the product of the segment lengths of one of them is equal to the product of the segment lengths of the other. Here, one of the chords is a diameter. That is special in that it is the bisector of any chord it crosses at right angles. That means ...
x·x = 2·6
x = √12 = 2√3
2. Secants from an external point have the same product of measures to the near and far intersection points with the circle. Since there is only one intersection point with a tangent, the near and far lengths are the same.
4·(4+8) = x·x
x = √48 = 4√3