+26367 Question One:
D is a point on \(\mathbf{\overline{AB}}\) such that \(\mathbf{AD = \dfrac13 AB}\).
If \(\mathbf{[ABC] = 12}\), what is \(\mathbf{[DBC]}\)?
\(\begin{array}{|lrcll|} \hline & [ABC] &=& \dfrac{AB\cdot BC\cdot \sin(B)}{2} \qquad (1) \\\\ & [DBC] &=& \dfrac{DB\cdot BC\cdot \sin(B)}{2} \qquad (2) \\\\ \hline \dfrac{(1)}{(2)} : & \dfrac{[ABC]}{[DBC]} &=& \dfrac{\dfrac{AB\cdot BC\cdot \sin(B)}{2} } {\dfrac{DB\cdot BC\cdot \sin(B)}{2}} \\\\ & \dfrac{[ABC]}{[DBC]} &=& \dfrac{AB} {DB} \\\\ & \dfrac{[DBC]}{[ABC]} &=& \dfrac{DB}{AB} \\\\ & [DBC] &=& \dfrac{DB}{AB}[ABC] \quad | \quad DB = \dfrac23 AB \\\\ & [DBC] &=& \dfrac{\dfrac23 AB}{AB}[ABC] \\\\ & [DBC] &=& \dfrac23[ABC] \quad | \quad [ABC] = 12 \\\\ & [DBC] &=& \dfrac23\cdot 12 \\\\ & \mathbf{[DBC]} & \mathbf{=} & \mathbf{8} \\ \hline \end{array}\)
