### Look Before You Leap

The diagonals of a square meet at O. The bisector of angle OAB meets BO and BC at N and P respectively. The length of NO is 24. How long is PC?

### Triangle Midpoints

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

### Trapezium Four

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

# Points in Pairs

##### Stage: 4 Challenge Level:

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