For a circle with center point P, cord XY is the

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For a circle with center point P, cord XY is the [#permalink]

08 Feb 2012, 18:44

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For a circle with center point P, cord XY is the perpendicular bisector of radius AP (A is a point on the edge of the circle). What is the length of cord XY?

(1) The circumference of circle P is twice the area of circle P.
(2) The length of Arc XAY = \frac{2\pi}{3}.

How come the answer is D? I have drawn these pictures as they were not provided with the questions. Even though with my guess work I have selected A which is incorrect. Can someone please let me know how to solve this? Also, I understand this will include a concept of 30-60-90 degree triangle - any idea which angles to assign 30 and 60 degrees?

Any idea how can I move this post to DS forum? My sincere apologies.

[Reveal] Spoiler: OA

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Re: Length of a Chord [#permalink]

08 Feb 2012, 19:33

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For a circle with center point P, cord XY is the perpendicular bisector of radius AP (A is a point on the edge of the circle). What is the length of cord XY?

From the diagram and the stem: AZ=ZP=r/2. In a right triangle ZPX ratio of ZP to XP is 1:2, hence ZPX is a 30-60-90 right triangle where the sides are in ratio: 1:\sqrt{3}:2. The longest leg is ZX which corresponds with \sqrt{3} and is opposite to 60 degrees angle. Thus <XPY=60+60=120

(1) The circumference of circle P is twice the area of circle P --> 2\pi{r}=2*\pi{r^2} --> r=1 --> XZ=\frac{\sqrt{3}}{2} --> XY=2*XZ=\sqrt{3}. Sufficient.

(2) The length of Arc XAY = 2pi/3 --> \frac{2\pi}{3}=\frac{120}{360}*2\pi{r} --> r=1, the same as above. Sufficient.

Answer: D.
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Director

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Re: For a circle with center point P, cord XY is the [#permalink]

11 Feb 2012, 12:29

Sorry Bunuel - in your explanation, how come longest leg be ZX? I think it should be XP because that's opposite to 90 degree angle. Also, do you mind telling me how did you find out which side will correspond to 60 degree and 30 degree angle?
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Re: For a circle with center point P, cord XY is the [#permalink]

11 Feb 2012, 12:45

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enigma123 wrote:

XP is hypotenuse, which obviously is the longest side but the longest leg is ZX (so the second longest side).

In a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio 1 : \sqrt{3}: 2. Notice that the smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°). Since the ratio of the leg ZP to the hypotenuse XP is 1:2, then ZP (the shortest side) corresponds to 1 and thus is the opposite of the smallest angle 30°, which means that another leg ZX corresponds to \sqrt{3}.

Hope it's clear.
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Re: For a circle with center point P, cord XY is the [#permalink]

11 Feb 2012, 13:52

Sorry Bunuel - still struggling. How did you get XZ = sqrt3/2? Apologies for been a pain.
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Re: For a circle with center point P, cord XY is the [#permalink]

11 Feb 2012, 14:35

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enigma123 wrote:

Sorry Bunuel - still struggling. How did you get XZ = sqrt3/2? Apologies for been a pain.

It's not a problem at all.

Since \frac{XZ}{XP}=\frac{\sqrt{3}}{2} (from 30°, 60°, and 90° right triangle ratio) and XP=r=1 then \frac{XZ}{1}=\frac{\sqrt{3}}{2} -->XZ=\frac{\sqrt{3}}{2}.

Hope it's clear.
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Re: For a circle with center point P, cord XY is the [#permalink]

10 Mar 2012, 00:20

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enigma123 wrote:

(2) The length of Arc XAY = 2p/3.

Why does it read 2p/3. Shouldn't it be 2pi/ 3 atleast, if you aren't putting the symbol for pi? That completely threw me off and I was left wondering, "does he mean 2(perimeter)/3? What does p stand for?"
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Re: For a circle with center point P, cord XY is the [#permalink]

10 Mar 2012, 00:29

budablasta wrote:

enigma123 wrote:

(2) The length of Arc XAY = 2p/3.

You could check the correct reading (\frac{2\pi}{3}) of the statement in my post above, though I edited the initial post to avoid confusion in others. Thanks for suggestion. +1.
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Re: For a circle with center point P, cord XY is the [#permalink] 10 Mar 2012, 00:29

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For a circle with center point P, cord XY is the

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