2020 CJC P1 Q11

A temporary isolation centre is built to manage the increasing number of COVID-19 cases. The roof takes the shape of a triangular prism. Points \((x,y,z)\) are defined relative to an origin, \(O\), with unit vectors \(\mathbf{i}\) along \(\overrightarrow{OA}\), \(\mathbf{j}\) along \(\overrightarrow{OC}\), and \(\mathbf{k}\) along \(\overrightarrow{OD}\) (see diagram). The coordinates of \(D\), \(E\), \(F\), \(G\) and \(I\) are \((0,0,3)\), \((4,0,3)\),\((4,12,4)\),\((0,12,4)\) and \((1,12,7)\) respectively. The units are measured in metres.

1. Find a cartiesian equation of the plane that contains the roof section \(EFIH\).[3]

It is given that the roof section \(DGIH\) is part of the plane with equation \(36x+y-12z=-36\).

2. Find a cartesian equation of the line that contains the roof ridge \(HI\).[3]

Steel cables are used to hold the isolation centre in place. Cables are laid in straight lines and the widths of cables can be neglected. It is given that cable \(1\) passess through \(F\) and \(D\) and cable \(2\) passes through \(G\) and \(M\), where \(M\) is the mid point of \(EF\).

A builder needs to locate the point \(J\) where both cables meet.

3. Find the coordinates of \(J\).[4]

To strengthen the structure, it is recommended that another steel cable should be extended from \(J\) to the closest point on the roof ridge \(HI\). The builder is left with \(3.2\) metres of steel cable.

4. Determine whether the remaining steel cable is long enough to connect \(J\) to the roof ridge \(HI\).[4]