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If A and C are points in a plane, C is the center of circle

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If A and C are points in a plane, C is the center of circle

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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 area of circle O is xπ.

OA A

Source: Princeton Review

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BTGmoderatorDC 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 area of circle O is xπ.

OA A

Source: Princeton Review
Given C is the center of the circle, point A is outside of the circle if AC = x > radius of the circle (r)

Let's take each statement one by one.

(1) The circumference of circle O is xπ.

Circumference of circle O = 2πr
=> 2πr = xπ

x = 2r

x > r. Sufficient.

(2) The area of circle O is xπ.

=> πr^2 = xπ

x = r^2

Case 1: Say r = 1/2, then r^2 = 1/4. We see that x < r. The answer is no.
Case 2: Say r = 2, then r^2 = 4. We see that x > r. The answer is yes.

Insufficient.

The correct answer: A

Hope this helps!

-Jay
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Quote:
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.

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