Final answer:
To construct a perpendicular line through point B, use a compass to draw arcs that intersect line a, then draw arcs from those points to find the perpendicular bisector and draw a straight line through B and this intersection.
Step-by-step explanation:
The best next step in the construction of a line that passes through point B and is perpendicular to line a involves using a compass and straightedge. Begin by placing the compass on point B, and draw an arc that intersects line a at two points. Keeping the compass at the same width, place it on one of the intersection points and draw another arc above line a.
Repeat this step with the other intersection point, creating two arcs that intersect above line a.
Draw a straight line from point B through this point of intersection of the arcs. This new line will be perpendicular to line a, intersecting it at a 90-degree angle.