Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
The above figure shows a sector of a circle. What is the area of the sector?
1) x = 120
2) AB=6√3
(We assume - as part of the definition of a sector a circle - that the "origin" of the angle x shown in the figure is the center of the circle.)
The variable R will denote the radius of the circle. All angles are measured in degrees.
\[? = \frac{{x\,}}{{360\,}}\left( {\pi {R^{\,2}}} \right)\]
\[x = 120\,\,\,\left( {{\text{both}}\,\,{\text{figures}}} \right)\]
\[AB = 6\sqrt 3 \,\,\,\left( {{\text{both}}\,\,{\text{figures}}} \right)\]
\[x = 90\,\,\,\mathop {\,\,\, \Rightarrow \,\,\,\,}\limits^{L\,,\,L\,,\,L\sqrt 2 } \,\,\,R\sqrt 2 = 6\sqrt 3 \,\,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,\frac{{\sqrt 2 }}{2}\,} \,\,\,\,\,R = 3\sqrt 6 \,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,?\,\,\, = \,\,\,\frac{1}{4}\,\left( {\pi \, \cdot 9 \cdot 6} \right)\]
\[x = 120\,\,\,\mathop \Rightarrow \limits_{\left( * \right)}^{30\,,\,60\,,\,90\,\,} \,\,\,\,?\,\,\, = \,\,\,\frac{1}{3}\,\left( {\pi \, \cdot 36} \right)\,\,\, = 12\pi \,\,\, \ne \,\,\,\,\frac{1}{4}\,\left( {\pi \, \cdot 9 \cdot 6} \right)\,\,\,\,\,\,\]
\[\left( * \right)\,\,30\,,\,60\,,\,90\,\,\,\, \Rightarrow \,\,\,\,\left\{ \begin{gathered}
\,L\sqrt 3 \,\,\, = \,\,\frac{{6\sqrt 3 }}{2} \hfill \\
\,2L = R \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,R = 6\]
\[\left( {1 + 2} \right)\,\,\,?\,\, = \,\,12\pi \,\,\,\left( {{\text{shown}}\,\,{\text{above}}} \right)\]
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
