Let's add all of the given information to the diagram. Note that ME and MN are both radii of ⊙ M. Also, AC and AR are both radii of ⊙ A.
According to the definition of a tangent, ME ⊥ ER and AR ⊥ ER. This must mean that ABER is a rectangle where BE=AR and BA=ER. With this information we can find the length of MB.
Examining the diagram, we can make out a right triangle where the hypotenuse and the shorter leg is known.
Using the Pythagorean Theorem, we can find the measure of AB and thereby also ER.
