AP=t，

OQ= 10-3t

Therefore, t= 10-3t.

Solution, t = 2.5s

From the beginning of the movement, after 2.5s, the quadrilateral AOQP is a parallelogram.

2. Let the quadrilateral AOQP become a right-angled trapezoid after ts, then there is OQ-AP=AO*cos60.

Namely: 10-3t-t=8*0.5.

The result is t =1.5s.

Therefore, during the movement of points P and Q, the quadrilateral AOQP can become a right-angled trapezoid.