2025 AHS PRELIMS P2 Q14

\(OACB\) is a quadrilateral. \(\overrightarrow{OA}=\mathbf a\), \(\overrightarrow{OB}=2\mathbf b\) and \(\overrightarrow{BC}=2\mathbf a+2\mathbf b\). \(BD:DA=2:1\).

1. Write each of the following in terms of \(\mathbf a\) and \(\mathbf b\),
   1. \(\overrightarrow{AB}\),[1]
   2. \(\overrightarrow{CA}\),[1]
   3. \(\overrightarrow{DO}\).[1]
2. Explain why \(\overrightarrow{DO}\) is parallel to \(\overrightarrow{BC}\).[1]
3. Show that \(\dfrac{\text{Area of triangle}\hspace{0.5em}OBD}{\text{Area of triangle}\hspace{0.5em}BCD}=\dfrac13\).[1]
4. Find
   1. \(\dfrac{\text{Area of triangle}\hspace{0.5em}BCD}{\text{Area of triangle}\hspace{0.5em}CAD}\),[1]
   2. \(\dfrac{\text{Area of triangle}\hspace{0.5em}OBD}{\text{Area of triangle}\hspace{0.5em}CAD}\).[1]