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805+ (Hard)|   Geometry|      
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For a circle with center point P cord XY is the perpendicular bisector 03 Mar 2014, 00:48
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lalania1
Hi Bunuel,

What formula have you applied to make statement 2 sufficient. Could you please explain. thanks!

(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.
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The formula is:

Arc length = \(2\pi{R}(\frac{C}{360})\), where C is the central angle of the arc in degrees.

Recall that \(2\pi{R}\) is the circumference of the whole circle, so the formula simply reduces this by the ratio of the arc angle to a full angle (360).

For more check Circles chapter of our Math Book: math-circles-87957.html

Hope it helps.
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For a circle with center point P cord XY is the perpendicular bisector 17 Apr 2014, 02:56
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Chord.PNG
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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HI Bunnel,

I am getting some doubts on this.

1. From statement 1 we can get that radius r = 1 . but how you are dividing r in r/2. We have given that XY is perpendicular bisector but this is not given it is dividing radius exactly in half. Also I can see diagrams are different in question and one you are referring.

Please clarify this. If in question also we have diagram as you have mentioned then I have no problem . I just want to know how you assumed to divide r in r/2

Thanks
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For a circle with center point P cord XY is the perpendicular bisector 17 Apr 2014, 03:23
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Bunuel



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.

HI Bunnel,

I am getting some doubts on this.

1. From statement 1 we can get that radius r = 1 . but how you are dividing r in r/2. We have given that XY is perpendicular bisector but this is not given it is dividing radius exactly in half. Also I can see diagrams are different in question and one you are referring.

Please clarify this. If in question also we have diagram as you have mentioned then I have no problem . I just want to know how you assumed to divide r in r/2

Thanks
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A perpendicular bisector is a line which cuts a line segment into two equal parts at 90°.

Thus XY is the perpendicular bisector of radius AP means that XY cuts AP into two equal parts.

Hope it's clear.
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For a circle with center point P cord XY is the perpendicular bisector 30 Oct 2015, 22:09
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since XY is perpendicular bisector of AP, we can draw a line from X to P and from Y to P, to form 2 30-60-90 triangles
why 30-60-90? because XP and YP is radius of the circle. AP is as well radius of the circle. Taking into consideration that XY bisects AP, we know for sure that:
from the intersection, say point Z, to P, we have 1/2 radius. we have 1 angle 90, and sides with proportions x-x sqrt 3 - 2x. this is a 30-60-90 triangle.

in order to find at least the radius.

from 1, we can deduct that r=2 - sufficient
from 2, we can deduct calculate the circumference, and get the radius - sufficient.

D
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For a circle with center point P cord XY is the perpendicular bisector 01 Nov 2015, 09:50
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Attachment:
a.png
a.png [ 4.33 KiB | Viewed 3904 times ]

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

(1) The circumference of circle P is twice the area of circle P.
(2) The length of Arc XAY = 2π3 .

If we let Z represent the intersection point between AP and XY, and ZP=r, XP=2r, XZ=sqrt(3)r, there is one variable (r), and 2 equations are given from the 2 conditions; there is high chance (D) will be our answer.
From condition 1, 2(phi)r=2(phi)r^2, r=1. This condition is sufficient,
From condition 2, angle AXP=120 deg, and the length of Arc XAY=2(phi)r(120/360)=2(phi)/3. r=1. This conditions is sufficient as well.
1)=2), so the answer becomes (D).
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For a circle with center point P cord XY is the perpendicular bisector 28 Nov 2017, 17:03
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Bunuel


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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Hi Bunuel,
Thanks for the amazing explanation. But how did you figure out that AZ=ZP? Do we have to assume it from the figure because it is nowhere mentioned in the question. For that reason I chose B which was incorrect.
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For a circle with center point P cord XY is the perpendicular bisector 28 Nov 2017, 20:32
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Bunuel


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.

Hi Bunuel,
Thanks for the amazing explanation. But how did you figure out that AZ=ZP? Do we have to assume it from the figure because it is nowhere mentioned in the question. For that reason I chose B which was incorrect.
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You should read the stem carefully. Check the highlighted part: bisector means that it cuts in half.
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For a circle with center point P cord XY is the perpendicular bisector 22 Jun 2018, 12:13
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Bunuel

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

Answer: D.

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Hi Bunuel,

Thanks for the explanation. Sorry this maybe a very basic question - how do we know that XZ=ZY?
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For a circle with center point P cord XY is the perpendicular bisector 22 Jun 2018, 22:42
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Bunuel

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

Answer: D.

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Attachment:
Chord.PNG

Hi Bunuel,

Thanks for the explanation. Sorry this maybe a very basic question - how do we know that XZ=ZY?
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Here is a proof:
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For a circle with center point P cord XY is the perpendicular bisector 04 Aug 2021, 14:06
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Bunuel bb, other moderators - need help here
In 2, how do we know r =1?
From the formula you used for arc length, we just get theta * r = 120.

I am having a hard time understanding how can we assume r = 1 here? it can be the angle subtended is 60 and the radius =2.

Thanks for the help
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For a circle with center point P cord XY is the perpendicular bisector 10 Aug 2021, 00:39
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Saby1098
Bunuel bb, other moderators - need help here
In 2, how do we know r =1?
From the formula you used for arc length, we just get theta * r = 120.

I am having a hard time understanding how can we assume r = 1 here? it can be the angle subtended is 60 and the radius =2.

Thanks for the help
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(2) says: "The length of Arc XAY = \(\frac{2\pi}{3}\)".

Arc XAY is 120 degrees, so its length is 1/3rd (120/360 = 1/3) of the whole circumference, which is \(2\pi{r}\). Thus, \(\frac{2\pi}{3}=\frac{120}{360}*2\pi{r}\). Solving this gives r = 1.
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For a circle with center point P cord XY is the perpendicular bisector 20 May 2022, 09:04
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Can someone please explain that why 2πr is = 2*πr^2 ? .logic might be little bit behind but I am new with circle so please give hands to me here.

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For a circle with center point P cord XY is the perpendicular bisector 16 Nov 2022, 12:44
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From the diagram and the stem: AZ=ZP=r/2. How?
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For a circle with center point P cord XY is the perpendicular bisector 17 Nov 2022, 02:30
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From the diagram and the stem: AZ=ZP=r/2. How?
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Have you read the whole discussion ? I think you must have missed this post.
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