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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?

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

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Weekly Problem 9 - 2006

weekly problem 9-2006

Points in Pairs

Stage: 4 Challenge Level: Challenge Level:1

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