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.
(i)
Explain why $p=2$.
[1]
(ii)
Find the Cartesian equation of plane $ABFE$.
[3]
(iii)
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.
(iv)
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 $OAED$.
(v)
Using your answer in part (iv) or otherwise, find a vector equation representing the straight cable.
[3]
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