Question

An architect plans to make a drawing of the room of a house. The segment LM represents the ceiling of the room. He wants to construct a line passing through Q and perpendicular to side LM to represent a wall of the room. He uses a straightedge and compass to complete some steps of the construction, as shown below:

Answer 1

A line segment LM is drawn. Two arcs equidistant from L on line LM are drawn at points P and T. A point Q is just above L. A compass is fixed at point Q and is shown making an arc at T. 
You draw arcs centered at P and T, such that they intersect above and below L. Note that the arcs must have radius greater than LT=LP. 

I get the feeling that Q is where the arcs intersect above L, so the intersection below L (at, say, S) means that QS is perpendicular to LM. 

Incidentally, constructing LM does not really fit the bill to start with, since either P or T must be beyond the line segment. What it should say is that points L and M are marked on a line extending beyond LM.