2019 TMJC Promo Q10

The diagram shows a structure with a horizontal square base \(OABC\) and a horizontal square top \(DEFG\), where \(OA=2\)m and \(DE=6\)m. The vertical height of the structure is \(8\) m. Each sloping face of the structure is a trapezium, and the edges \(AE\), \(BF\), \(CG\) and \(OD\) are of equal length.

The point \(O\) is taken as the origin and perpendicular unit vectors \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) are such that \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OA\) and \(OC\) respectively. The coordinates of points \(D\), \(E\), \(F\) and \(G\) can be expressed as \(\left( -p,\,\,-p,\,\,8 \right)\), \(\left( 2+p,\,\,-p,\,\,8 \right)\), \(\left( 2+p,\,\,2+p,\,\,8 \right)\), and \(\left( -p,\,\,2+p,\,\,8 \right)\) respectively.

1. Explain why \(p=2\).[1]
2. Find the Cartesian equation of plane \(ABFE\).[3]
3. Given that the Cartesian equation of plane \(OAED\) is \(4y+z=0\), determine the obtuse angle between planes \(OAED\) and \(ABFE\).[2]

A rod connects point \(M\), the midpoint of edge \(FG\), to a point \(N\) on the plane \(OAED\) such that \(MN\) is made as small as possible.

4. Find the coordinates of \(~N\) and the exact minimum length of the rod.[5]

The structure sits on the ground and is secured by a straight cable connecting a point on the ground to the point \(D\). This cable coincides with the line of reflection of the line segment \(DM\) in the plane \(OAED\).

5. Using your answer in part (iv) or otherwise, find a vector equation representing the straight cable.[3]