So for QC to be a perpendicular bisector, it needs to make BC=CD lets label BC as X. BC+CD=BD. Because BC=CD, X+X=BD or 2X=24. Thus X=12. Thus BC=12. Now PB is a perpendicular bisector of AC so PB makes a right angle with AC at point B. PC=13(given) so because we have a right triangle, we can use the hypotenuse theorem so PC^2=BC^2+PB^2 or 13^2=12^2+PB^2 so 25=PB^2 so PB=5
Hope this helps!