2019 ASRJC P1 Q9

The diagram above shows an object with \(O\) at the centre of its rectangular base \(ABCD\) where \(AB=8\)cm and \(BC=4\)cm. The top side of the object, \(EFGH\) is a square with side \(2\)cm long and is parallel to the base. The centre of the top side is vertically above \(O\) at a height \(h\) cm.

1. Show that the equation of the line \(BG\) may be expressed as \(r=\left( \begin{matrix}4 \\-2 \\ 0 \end{matrix} \right)+t\left( \begin{matrix}-3 \\ 1 \\ h \end{matrix} \right)\), where \(t\) is a parameter.[1]
2. Find the sine of the angle between the line \(BG\) and the rectangular base \(ABCD\) in terms of \(h\).[2]

It is given that \(h=6\).

3. Find the cartesian equation of the plane \(BCFG\).[3]
4. Find the shortest distance from the point \(A\) to the plane \(BCFG\).[2]
5. The line \(l\), which passes through the point \(A\), is parallel to the normal of plane \(BCFG\). Given that, the line \(l\) intersects the plane \(BCFG\) at a point \(M\), use your answer in part (iv) to find the shortest distance from point \(M\) to the rectangular base \(ABCD\).[2]