We had a couple of correct solutions (500/48 or 10.4 to 3sf) using the Cosine Rule, and one solution which used the much simpler method of similar triangles:

I noticed that since $OP'=100/OP$ and $OQ'=100/OQ$ the ratio $\frac{OP}{OQ}$, is \frac{OQ'}{OP'} and so the triangles $OPQ$ and $OQ'P'$ are similar.

So in this question $OP'=10/8$,and since the ratio between lengths of sides is the same by similar triangles: $$ \frac{P'Q'}{OP'}=\frac{PQ}{OQ}$$ So: $$P'Q'=\frac{100}{6}\times\frac{5}{8}=\frac{500}{48}$$.