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In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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102 00:00

Difficulty:   85% (hard)

Question Stats: 59% (02:11) correct 41% (02:04) wrong based on 2133 sessions

### HideShow timer Statistics In the figure above, points A, B, C, D, and E lie on a line. A is on both circles, B is the center of the smaller circle, C is the center of the larger circle, D is on the smaller circle, and E is on the larger circle. What is the area of the region inside the larger circle and outside the smaller circle?

(1) AB = 3 and BC = 2
(2) CD = 1 and DE = 4

Attachment: Two circles.png [ 5.47 KiB | Viewed 38422 times ]

Originally posted by Walkabout on 14 Dec 2012, 02:31.
Last edited by Bunuel on 05 Feb 2019, 22:15, edited 1 time in total.
Renamed the topic.
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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19
20 In the figure above, points A, B, C, D, and E lie on a line. A is on both circles, B is the center of the smaller circle, C is the center of the larger circle, D is on the smaller circle, and E is on the larger circle. What is the area of the region inside the larger circle and outside the smaller circle?

We need to find the value of $$\pi{R^2}-\pi{r^2}$$, where R is the radius of the larger circle and r is the radius of the smaller circle. So, bascally we need to find R and r.

(1) AB = 3 and BC = 2 --> $$AB=r=3$$ and $$AB+BC=AC=5=R$$. Sufficient.

(2) CD = 1 and DE = 4 --> $$CD+DE=CE=R=5$$. Next, $$AD+DE=AE=DIAMETER=2R=10$$, and since $$AD=2r$$, then $$2r+DE=10$$ --> $$2r+4=10$$ --> $$r=3$$. Sufficient.

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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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2
From the figure we know that:
B is the center of the smaller circle
C is the center of the larger circle
We have to find out area of region i.e area obtained when area of the smaller circle removed from area of the larger circle.

Statement 1

AB=3 and BC= 2

from this statement we know radius of the smaller circle i.e 3(AB) and radius of the larger circle as well $$3+2=5 (AB+BC)$$
using radius of both the circle we can find the area of each circle and hence the required area.
Sufficient

Statement 2

CD=1 and DE=4

from this statement we know that radius of the larger circle is $$1+4=5 (CD+DE)$$
now for the smaller circle,we know that AE=10 and we know DE=4. therefore diameter of the smaller circle is $$10(AE)-4(DE)=6 (AC)$$
therefore radius of the smaller circle $$= 3 (\frac{AC}{2})$$. using radius of both the circle we can find the area of each circle and hence the required area.
Sufficient

Originally posted by AayushGMAT on 18 Sep 2014, 13:11.
Last edited by AayushGMAT on 26 Oct 2015, 00:23, edited 1 time in total.
##### General Discussion
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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Kudos for the solution. I tried to solve the problem twice, but could not get it. I am hoping to start getting solutions to such problems with more practice Manager  B
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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Bunuel - As we know CE=AC=5 so AD=2r=AC+CD=6, can this be an alternative way to confirm that statement#2 is sufficient?
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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Dienekes wrote:
Bunuel - As we know CE=AC=5 so AD=2r=AC+CD=6, can this be an alternative way to confirm that statement#2 is sufficient?

______________
Yes, that's correct.
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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Bunuel, are there any similar problems to this one?
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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Ergenekon wrote:
Bunuel, are there any similar problems to this one?

We can consider this question as shaded region problem. Check other Shaded Region Problems in our Special Questions Directory.
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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In this question Why are we considering C towards left of D its not stated in the question that its in left of D its shown but not stated. C is given that its lie on the same line and its center of larger circle, But what if the C is towards right of D. Then we can not solve this question using only statement 2.

Should the diagram be considered as concrete not flexible?
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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Parth Monga wrote:
In this question Why are we considering C towards left of D its not stated in the question that its in left of D its shown but not stated. C is given that its lie on the same line and its center of larger circle, But what if the C is towards right of D. Then we can not solve this question using only statement 2.

Should the diagram be considered as concrete not flexible?

OG13, page 272:
A figure accompanying a data sufficiency problem will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).
Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.
You may assume that the positions of points, angles, regions, and so forth exist in the order shown and that angle measures are greater than zero degrees.
All figures lie in a plane unless otherwise indicated.

OG13, page 150:
Figures: A figure accompanying a problem solving question is intended to provide information useful in solving the problem. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.

Hope it helps.
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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Bunuel wrote:
Parth Monga wrote:
In this question Why are we considering C towards left of D its not stated in the question that its in left of D its shown but not stated. C is given that its lie on the same line and its center of larger circle, But what if the C is towards right of D. Then we can not solve this question using only statement 2.

Should the diagram be considered as concrete not flexible?

OG13, page 272:
A figure accompanying a data sufficiency problem will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).
Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.
You may assume that the positions of points, angles, regions, and so forth exist in the order shown and that angle measures are greater than zero degrees.
All figures lie in a plane unless otherwise indicated.

OG13, page 150:
Figures: A figure accompanying a problem solving question is intended to provide information useful in solving the problem. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.

Hope it helps.

This a lot Bunuel that solves my concern with this question. Thanks a lot for your help.
P.S. the content provided in your signature is quite good and helpful. Really appreciate your work and effort.
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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Attachment:
Two circles.png
In the figure above, points A, B, C, D, and E lie on a line. A is on both circles, B is the center of the smaller circle, C is the center of the larger circle, D is on the smaller circle, and E is on the larger circle. What is the area of the region inside the larger circle and outside the smaller circle?

(1) AB = 3 and BC = 2
(2) CD = 1 and DE = 4

A - clearly sufficient
B- cd+de= radius so smaller circle can also be found if you use it in opposite direcetion - sufficient
OA:D
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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I have one question, how do we know CE is the diameter. The problem never says it is?
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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Turkish wrote:
I have one question, how do we know CE is the diameter. The problem never says it is?

CE is the radius of the large circle because C is its center and E is on its circumference.
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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Bunuel wrote:
Turkish wrote:
I have one question, how do we know CE is the diameter. The problem never says it is?

CE is the radius of the large circle because C is its center and E is on its circumference.

Sorry Bunuel may be I am not getting this. In the problem it never say E is one the circumference, It says E is on larger circle. Should we believe in what we see,even if its not mentioned?
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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Turkish wrote:
Bunuel wrote:
Turkish wrote:
I have one question, how do we know CE is the diameter. The problem never says it is?

CE is the radius of the large circle because C is its center and E is on its circumference.

Sorry Bunuel may be I am not getting this. In the problem it never say E is one the circumference, It says E is on larger circle. Should we believe in what we see,even if its not mentioned?

E is on the larger circle means that E is on the circumference.
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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1
We need to find AREA (BIG)-(SMALL) --> P*R^2 - P*R^2
1) AB=3, BC=2, R(big)=3+2=5, R(small)=3 Sufficient
2) CD=1, DE=4, Diameter of the big circle = CD+DE=5, Diameter of the small circle = AC+CD=5+1=6, Radius =3 Sufficient
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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In this question what we need to decide is that area of both the circles can be calculated or not
Area of circle = π r^2
from (1):radius of inner circle = AB=3(given);Radius of outer circle = AC=AB+BC=3+2=5 Both area can be calculated. Hence Sufficient
radius of inner circle=>AB=BD(since b is the center and D is a point on inner circle)=AD/2=6/2=3. Both area can be calculated. Hence Sufficient

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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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Bunuel wrote: In the figure above, points A, B, C, D, and E lie on a line. A is on both circles, B is the center of the smaller circle, C is the center of the larger circle, 0 is on the smaller circle, and E is on the larger circle. What is the area of the region inside the larger circle and outside the smaller circle?

We need to find the value of $$\pi{R^2}-\pi{r^2}$$, where R is the radius of the larger circle and r is the radius of the smaller circle. So, bascally we need to find R and r.

(1) AB = 3 and BC = 2 --> $$AB=r=3$$ and $$AB+BC=AC=5=R$$. Sufficient.

(2) CD = 1 and DE = 4 --> $$CD+DE=CE=R=5$$. Next, $$AD+DE=AE=DIAMETER=2R=10$$, and since $$AD=2r$$, then $$2r+DE=10$$ --> $$2r+4=10$$ --> $$r=3$$. Sufficient.

i dont get is D in the center ? Math Expert V
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Re: In the figure above, points A, B, C, D, and E lie on a line. A is on  [#permalink]

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dave13 wrote:
Bunuel wrote: In the figure above, points A, B, C, D, and E lie on a line. A is on both circles, B is the center of the smaller circle, C is the center of the larger circle, D is on the smaller circle, and E is on the larger circle. What is the area of the region inside the larger circle and outside the smaller circle?

We need to find the value of $$\pi{R^2}-\pi{r^2}$$, where R is the radius of the larger circle and r is the radius of the smaller circle. So, bascally we need to find R and r.

(1) AB = 3 and BC = 2 --> $$AB=r=3$$ and $$AB+BC=AC=5=R$$. Sufficient.

(2) CD = 1 and DE = 4 --> $$CD+DE=CE=R=5$$. Next, $$AD+DE=AE=DIAMETER=2R=10$$, and since $$AD=2r$$, then $$2r+DE=10$$ --> $$2r+4=10$$ --> $$r=3$$. Sufficient.

i dont get is D in the center ? No, D is not the centre of any circle in the image. Check the highlighted parts.
_________________ Re: In the figure above, points A, B, C, D, and E lie on a line. A is on   [#permalink] 23 Aug 2018, 19:51

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