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Find the length

Source: Romanian District Olympiad 2014, Grade 8, P1

June 15, 2014
analytic geometrygeometry proposedgeometry

Problem Statement

In the right parallelopiped ABCDA′B′C′D′ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}, with AB=123AB=12\sqrt{3} cm and AA′=18AA^{\prime}=18 cm, we consider the points P∈AA′P\in AA^{\prime} and N∈A′B′N\in A^{\prime}B^{\prime} such that A′N=3B′NA^{\prime}N=3B^{\prime}N. Determine the length of the line segment APAP such that for any position of the point M∈BCM\in BC, the triangle MNPMNP is right angled at NN.
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