The Answer states: Using statements 1 and 2, we know that AC is the perpendicular bisector of BD. This means that triangle BAD is an isosceles triangle so side AB must have a length of 5 (the same length as side AD).

How do you know that because a line is a perpendicular bisector splitting the base of a triangle into two equal parts, the initial triangle that is split into two is an isosceles?

(sorry I don't know how to put the question up on this post)

Mathematical proof. If you split the base into two identical pieces via the perpendicular bisector, then you also split that angle (at the vertex of the triangle) into two identical angles. Two equal angles in a triangle means the two corresponding sides are also equal in length.