We have to show that OP II QR, i.e., that both
have the same slope. 

mOP =  =  = 

mQR =  =  = 

Their slopes are equal, so OPQR is either a trapezoid or a parallelogram.

Now let's show that PQ = RO

DPQ =  =  = =  =  

DRO =  =  =  =  = 

You might think this is enough to show that that OPQR is an isosceles
trapezoid since one pair of opposite sides are parallel and the other 
pair of opposite sides are equal in length.

However, that could be said about a rectangle.  We must rule out
a rectangle. (Yes we can look and see that it's not a rectangle,
but "looking and seeing" is not acceptable. The easiest way to do 
this is to rule out a right angle using slopes. We will show that
OP is not perpendicular to RO

We have already found the slope of OP as 1/3. So we will rule out
OPQR as being a rectangle by showing that the slope of RO is not
the "negative reciprocal" of 1/3.  That is to show that the slope of
RO is not -3.

mRO =  =  = 

 is not equal to  so we have ruled out a rectangle.

So OPQR must be an isosceles trapezoid.

Edwin