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 OQ' 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:

$P' Q' = \frac{100}{6} \times \frac{5}{8} = \frac{500}{48}$

.