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
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:
P¢Q¢
OP¢
 = PQ
OQ
So:
P¢Q¢= 100
6
 × 5
8
 = 500
48
.