PRMO 2013 question 12

 Paper


Let \(ABC\) be an equilateral triangle of side length \(s\). Let \(P\) and \(S\) be points on \(AB\) and \(AC\), respectively, and let \(Q\) and \(R\) be points on \(BC\) such that \(PQRS\) is a rectangle. If \[ PQ = \sqrt3\,PS \quad\text{and}\quad \text{Area}(PQRS)=28\sqrt3, \] what is the length of \(PC\)?

Solution:
First find that PS = \(2\sqrt7\) and PQ = \(2\sqrt21\). Now APS and ABC are similar so APS is also equilateral. Hence AP = PS = AS = \(2\sqrt7\).

Let's say SC = x, then RC = x*cos(60 deg) = x/2 and SR = x * sqrt(3)/2
So SC = 4*sqrt(7)
And RC = 2*sqrt(7)
So each side of the triangle is 6*sqrt(7).
QC = QR + RC = 2*sqrt(7) + 2*sqrt(7)
Now PC = Sqrt(PQ^2 + QC^2) = 14

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