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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  [#permalink]

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27 May 2012, 09:36
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Question Stats:

51% (01:51) correct 49% (01:47) wrong based on 264 sessions

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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.
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Re: If A and C are points in a plane, C is the center of circle  [#permalink]

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28 May 2012, 03:49
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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 ?

Notice that point A will be outside the circle if the radius of the circle is less than x (the distance between the center of the circle and point A). So, the question basically asks whether $$r<x$$.

(1) The circumference of circle O is xπ --> $$2\pi{r}=x\pi$$ --> $$2r=x$$ --> $$x>r$$. Sufficient.
(2) The area of circle O is xπ --> $$\pi{r^2}=x\pi$$ --> $$x=r^2$$. If $$x=4$$ and $$r=2$$ then the answer is YES but if $$x=\frac{1}{4}$$ and $$r=\frac{1}{2}$$ then the answer is NO. Not sufficient.

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Re: If A and C are points in a plane, C is the center of circle  [#permalink]

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27 May 2012, 19:01
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Perimeter of circle is 2*pi*r. From first statement we can find out that x is the diameter of circle so point A has to be outside of the circle.

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Re: If A and C are points in a plane, C is the center of circle  [#permalink]

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24 Jul 2016, 06:03
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 area of circle O is xπ.

(1) 2πr=xπ
2r=x-----Dia of circle.
if c is centre and AC=x then point A must lie outside of circle.
(2)πr^2=xπ
r^2=x
x=2 it means AC=2 and radius of circle is Sq.root2 then A may lie inside or outside the circle since dia. of circle is 2*sq.root2....insuff...
Ans A
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If A and C are points in a plane, C is the center of circle  [#permalink]

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11 Nov 2017, 20:10
Bunuel 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 ?

Notice that point A will be outside the circle if the radius of the circle is less than x (the distance between the center of the circle and point A). So, the question basically asks whether $$r<x$$.

(1) The circumference of circle O is xπ --> $$2\pi{r}=x\pi$$ --> $$2r=x$$ --> $$x>r$$. Sufficient.
(2) The area of circle O is xπ --> $$\pi{r^2}=x\pi$$ --> $$x=r^2$$. If $$x=4$$ and $$r=2$$ then the answer is YES but if $$x=\frac{1}{4}$$ and $$r=\frac{1}{2}$$ then the answer is NO. Not sufficient.

Hi Bunuel

Do we have point circle concept in Gmat?
if so, From (1) 2r=x=0 could also be a case and so point lie on circle. And this may become insufficient.

From (2) It's anyways eliminated even without the concept of point circle

From (1) & (2) we still have two cases so It may lead to "E"

From OG:
This is how they defined Line segment: ( It was not mentioned if P & Q should be distinct.)
The part of the line from P to Q is called a line segment. P and Q are the endpoints of the segment. The notation PQ is used to denote line segment PQ and PQ is used to denote the length of the segment.

This is how circe is defined: (The set can also have just one element?)
A circle is a set of points in a plane that are all located the same distance from a fixed point.

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Re: If A and C are points in a plane, C is the center of circle  [#permalink]

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16 Sep 2018, 11:26
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.
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Re: If A and C are points in a plane, C is the center of circle   [#permalink] 16 Sep 2018, 11:26
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