macjas wrote:

If A and C are points in a plane, C is the center of circle O, and the length of line segment AC is x, does point A lie outside circle O ?

(1) The circumference of circle O is xπ.

(2) The numerical value of the area of circle O is equal to the numerical value of xπ.

Obs.:

xπ is measured in unit of length, while areas are measured in unit of length squared. That´s why we modified statement (2) accordingly.

\({\text{dist}}\left( {A,C} \right)\,\,\, = \,\,x\,\,\,\,\mathop > \limits^? \,\,\,r\)

\(\left( 1 \right)\,\,\,\,2\pi r = \pi x\,\,\,\,\mathop \Rightarrow \limits^{:\,\,\pi \,} \,\,\,\,\,x = 2r\,\,\mathop > \limits^{r\,\, > \,0} r\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle\)

\(\left( 2 \right)\,\,\,\pi {r^2} = x\pi \,\,\,\,\mathop \Rightarrow \limits^{:\,\,\pi \,} \,\,\,\,\,x = {r^2}\,\,\,\left\{ \begin{gathered}

\,{\text{Take}}\,\,r = 2\,\,\,\,\, \Rightarrow \,\,\,\,\,x = 4 > 2\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\

\,{\text{Take}}\,\,r = 0.5\,\,\,\,\, \Rightarrow \,\,\,\,\,x = 0.25 < 0.5\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{N}}{\text{O}}} \right\rangle \hfill \\

\end{gathered} \right.\)

This solution follows the notations and rationale taught in the GMATH method.

Regards,

Fabio.

_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)

Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!