Draw two circles through B with their centres on the given line. The line from B to the other point of intersection of the two circles does the job. If you want the segment to explicitly start on the line, then find the intersection of the two lines drawn and draw a line segment from there to B.
Bizarre. It didn't work for me earlier, but it does now. Maybe I didn't get the intersection point D correct?
Sorry for the noise, and thanks for creating this great activity!
Draw two circles through B with their centres on the given line. The line from B to the other point of intersection of the two circles does the job. If you want the segment to explicitly start on the line, then find the intersection of the two lines drawn and draw a line segment from there to B.