In the following figure, AF is the perpendicular bisector of BD, AD = 30, and DF = 12. Find AB + BF.

Since AF is the perpendicular bisector both angle(AFD) and angle(AFB) are right angles; 

also, BF is congruent to DF.

Since AF is congruent to AF, triangle(ABF) is congruent to triangle(ADF)   ---   side-angle-side.

Therefore, AD = AB = 30 and DF = BF = 12.

 

You can now calculate AB + BF.