Author 
Message 
TAGS:


Moderator
Joined: 18 Apr 2015
Posts: 5567
Followers: 89
Kudos [?]:
1117
[0], given: 5159

In the figure above, the diameter of the circle is 20 and th [#permalink]
19 May 2017, 06:38
Question Stats:
60% (01:54) correct
39% (02:06) wrong based on 48 sessions
Attachment:
#GREpracticequestion In the figure above, the diameter of the circle is 20.jpg [ 13.37 KiB  Viewed 379 times ]
In the figure above, the diameter of the circle is 20 and the sum of the area of the sectors CFAO and BGDO is \(80\pi\). What is the value of \(a + b + c + d\)? A) 144 B) 216 C) 240 D) 270 E) 288
_________________
Get the 2 FREE GREPrepclub Tests
Last edited by Carcass on 30 Jan 2019, 04:03, edited 7 times in total.
Corrected the Q




Intern
Joined: 14 May 2017
Posts: 3
Followers: 0
Kudos [?]:
4
[2]
, given: 2

Re: In the figure above [#permalink]
20 May 2017, 04:13
2
This post received KUDOS
Damm bro, I think the question looks a bit confusing at the beginning... What is missing is the information that the area of the circle WITHOUT the two triangles is 80. It´s not possible in a other way, since the radius is 10, the area is 100*pi.
Given that, we can make an approximation for a, b, c and d using circle segments. It is only an approximation because we assume the area of the triangle is the same as the area of the respective circle segment. This is fair enough for small triangles and since we do not need to calculate the value. We just have to come near to a provided solution.
So let us say the inner angles of the triangles are called e and f respectively. Since the total are is 100 pi and the area without the two triangles is 80 pi, the two triangles have the area 20 pi. And one triangle has the area of 10 pi, which is exactly 10% of the total area. Again, assuming the triangle is a circle segment, the inner angle (e and f) must be 360/10=36 degrees. Since a=b=c=d, a+b=18036 => a+b=144. Therefore a+b+c+d= 144*2=288.
This is the value provided as OA.



Intern
Joined: 08 Dec 2017
Posts: 40
Followers: 1
Kudos [?]:
38
[3]
, given: 70

Re: In the figure above [#permalink]
25 Jan 2018, 02:56
3
This post received KUDOS
D=20 so the r=10. So the circle is 100pi. Given that, the circle area without the two triangles is 80pi. So the area of the two triangles is 20pi. Now simply, considering a triangle as a sector of the circle we can use the formula (theta/360)=(Sector area/circle area). By solving, theta=36 degrees. So a sector creates an angle of 36 degrees in the center and leaving 144 degrees for other two angles of a triangle. So for two triangles' we can get 144+144=288 degrees.



Intern
Joined: 15 Mar 2018
Posts: 32
Followers: 0
Kudos [?]:
7
[1]
, given: 1

Re: In the figure above [#permalink]
03 Apr 2018, 07:26
1
This post received KUDOS
E is not the answer and no correct answer is given!
The question read the area of the circle without the two triangles is 80 pi. Not the area of the circle without the sector, both answers above are incorrect.
The area of the two triangles is 2 * ( 1/2 * r^2 * Sin ( Q)) where Q is the angle of the triangle's apex at the center of the circle.
So 20pi = r^2 * Sin (Q) .... Q = asin ( 0.2) = 11.537... thus a + b + c + d = 2* (18011.537) = 336.926.



Intern
Joined: 08 Mar 2018
Posts: 11
Followers: 0
Kudos [?]:
8
[1]
, given: 10

Re: In the figure above [#permalink]
03 Apr 2018, 13:25
1
This post received KUDOS
YMAkib wrote: D=20 so the r=10. So the circle is 100pi. Given that, the circle area without the two triangles is 80pi. So the area of the two triangles is 20pi. Now simply, considering a triangle as a sector of the circle we can use the formula (theta/360)=(Sector area/circle area). By solving, theta=36 degrees. So a sector creates an angle of 36 degrees in the center and leaving 144 degrees for other two angles of a triangle. So for two triangles' we can get 144+144=288 degrees. I am not getting 36 degrees for theta/360 = 20pi/100pi. Can you please explain how you got to 36? Thank you



Moderator
Joined: 18 Apr 2015
Posts: 5567
Followers: 89
Kudos [?]:
1117
[2]
, given: 5159

Re: In the figure above [#permalink]
03 Apr 2018, 13:38
2
This post received KUDOS
There are two versions of this question. I edited the question above changing is the shaded region that I suppose is the area inside the circle but OUTSIDE the two triangles. Now, the radius is 10 and the area is 100 \pi. The area of the shadow region is \(\frac{80}{100} = \frac{4}{5}\) of the circle and the rest is \(\frac{1}{5}\) of the circle. We do know that the central angle in a circle is always \(360\) and \(\frac{1}{5}\) of 360 is 72°. \(a + b + c + d + 72 = 360\) \(a + b + c + d = 288\)
_________________
Get the 2 FREE GREPrepclub Tests



Intern
Joined: 02 Jun 2018
Posts: 1
Followers: 0
Kudos [?]:
0
[0], given: 2

Re: In the figure above [#permalink]
03 Jun 2018, 23:09
Which area is the shaded one?



Moderator
Joined: 18 Apr 2015
Posts: 5567
Followers: 89
Kudos [?]:
1117
[0], given: 5159

Re: In the figure above [#permalink]
04 Jun 2018, 11:10
Unfortunately, the book reports this kind of graph. It should be the area inside the two triangles. regards
_________________
Get the 2 FREE GREPrepclub Tests



Manager
Joined: 02 Dec 2018
Posts: 74
Followers: 0
Kudos [?]:
7
[1]
, given: 60

Re: In the figure above [#permalink]
13 Dec 2018, 18:07
1
This post received KUDOS
Carcass wrote: Unfortunately, the book reports this kind of graph.
It should be the area inside the two triangles.
regards I would suggest to edit the original post indicating that.



Moderator
Joined: 18 Apr 2015
Posts: 5567
Followers: 89
Kudos [?]:
1117
[1]
, given: 5159

Re: In the figure above [#permalink]
14 Dec 2018, 03:32
1
This post received KUDOS



Moderator
Joined: 18 Apr 2015
Posts: 5567
Followers: 89
Kudos [?]:
1117
[0], given: 5159

Re: In the figure above, the diameter of the circle is 20 and th [#permalink]
26 Dec 2018, 11:27
This was one of the most controversial questions due to the poor quality of the geometry figure. Now I have updated with a good one figure. Regards
_________________
Get the 2 FREE GREPrepclub Tests



Intern
Joined: 14 Jan 2019
Posts: 31
Followers: 0
Kudos [?]:
5
[1]
, given: 5

Re: In the figure above, the diameter of the circle is 20 and th [#permalink]
28 Jan 2019, 13:49
1
This post received KUDOS
Area of Sector: x/360 * πR^2 Two Sector was shaded so each sector = 80π/2 = 40π
x/360 * πR^2 = 40π x=144
a+b=c+d=144
Since R=D/2=10 a=b=c=d
a=b=c=d=72
a+b+c+d=72*4=288



Supreme Moderator
Joined: 01 Nov 2017
Posts: 349
Followers: 4
Kudos [?]:
98
[0], given: 2

Re: In the figure above, the diameter of the circle is 20 and th [#permalink]
28 Jan 2019, 19:53
Carcass wrote: This was one of the most controversial questions due to the poor quality of the geometry figure.
Now I have updated with a good one figure.
Regards Hi carcass, The question or the figure is still flawed. The answer would be 288 ONLY when you are looking at the area of 80 pi for all the area outside the two sector and not the area outside the triangle.
_________________
Some useful Theory. 1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressionsarithmeticgeometricandharmonic11574.html#p27048 2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effectsofarithmeticoperationsonfractions11573.html?sid=d570445335a783891cd4d48a17db9825 3. Remainders : https://greprepclub.com/forum/remainderswhatyoushouldknow11524.html 4. Number properties : https://greprepclub.com/forum/numberpropertyallyourequire11518.html 5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolutemodulusabetterunderstanding11281.html



Moderator
Joined: 18 Apr 2015
Posts: 5567
Followers: 89
Kudos [?]:
1117
[0], given: 5159

Re: In the figure above, the diameter of the circle is 20 and th [#permalink]
30 Jan 2019, 04:09
Thanks, Cethan. However, the figure, the question, and the answer choices are reported as they are in Barron's book without typos or else. Faithfully. Regards
_________________
Get the 2 FREE GREPrepclub Tests



Supreme Moderator
Joined: 01 Nov 2017
Posts: 349
Followers: 4
Kudos [?]:
98
[0], given: 2

Re: In the figure above, the diameter of the circle is 20 and th [#permalink]
30 Jan 2019, 04:35
Carcass wrote: Thanks, Cethan.
However, the figure, the question, and the answer choices are reported as they are in Barron's book without typos or else. Faithfully.
Regards I have corrected the figure. Hopefully it is clear and no more ambiguous as given by Barron's. Thanks
_________________
Some useful Theory. 1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressionsarithmeticgeometricandharmonic11574.html#p27048 2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effectsofarithmeticoperationsonfractions11573.html?sid=d570445335a783891cd4d48a17db9825 3. Remainders : https://greprepclub.com/forum/remainderswhatyoushouldknow11524.html 4. Number properties : https://greprepclub.com/forum/numberpropertyallyourequire11518.html 5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolutemodulusabetterunderstanding11281.html



Director
Joined: 20 Apr 2016
Posts: 817
WE: Engineering (Energy and Utilities)
Followers: 9
Kudos [?]:
588
[1]
, given: 113

Re: In the figure above, the diameter of the circle is 20 and th [#permalink]
02 Feb 2019, 11:23
1
This post received KUDOS
Carcass wrote: This was one of the most controversial questions due to the poor quality of the geometry figure.
Now I have updated with a good one figure.
Regards @Carcass Plz can u look into the diagram attached When the area of triangle is said to \(10\pi\) are we actually considering the the area of the arc AB and arc AC? I didn't really understood the reasoning However, this is how I solved, @Carcass kindly provide ur valuable feedback::Let first find the angle COA and angle DOB i.e. \(\frac{{\angle COA}}{360} = \frac{{40\pi}}{{100\pi}}\) (each shaded area is 40 pi and the area of the circle is 100 pi) or \(\angle COA = 144\) Now \(\angle COD = 180 144 =36\) (DA is the a straight line and diameter of the circle) Similarly it can be proved \(\angle AOB = 36\) Hence for the two \(\triangle\)'s combined it can be written as \(a + b + 36 + c + d + 36 =360\) or \(a + b + c + d = 288\)
Attachments
#GREpracticequestion In the figure above, the diameter of the circle is 20.jpg [ 15.74 KiB  Viewed 285 times ]
_________________
If you found this post useful, please let me know by pressing the Kudos Button
Rules for Posting https://greprepclub.com/forum/rulesfor ... 1083.html



Moderator
Joined: 18 Apr 2015
Posts: 5567
Followers: 89
Kudos [?]:
1117
[1]
, given: 5159

Re: In the figure above, the diameter of the circle is 20 and th [#permalink]
03 Feb 2019, 01:22
1
This post received KUDOS
This question clearly has a bit of a flaw in its wording. Certainly is that the letters to pinpoint the shaded area by Chetan came in handy. However, I always try to look at the bright side of the story and stretch my mental muscles. If we look at the original graph as it has been provided by the book and using logic, the solution will be easier. Of course, your solution is elegant and fine, though the arcs way to solve it I guess is a bit cumbersome. I always try to go down the road straight. Attachment:
shot39.jpg [ 18.31 KiB  Viewed 245 times ]
Now, few easy steps  the stem says the diameter is 20 so the radius must be 10.
 If the radius is 10 the entire area of the circle is \(100 \pi\)
At this point, the stem also says us that the shaded region is \(80 \pi\). At the time I didn't have the right graph and it looks like the following Attachment:
shot40.jpg [ 11.03 KiB  Viewed 245 times ]
Even having this graph, is impossible to think that the shaded area is inside the isosceles triangles. it must be the outside of them. As it turns out, it is the rest of the circle. The total area is \(100 \pi\) so having \(80 \pi\) the latter represents 80% of the entire area. From this, the area of the triangles is \(\frac{20}{100} = \frac{1}{5}\) of the entire area. From this, we do know that inside an isosceles triangle, two sides are equal. The triangles inside the circle have the two major sides = 10. Therefore, the four angles a  b  c  d are the biggest. However, which is their measure ?? We do know that the sum of the angles is 180° if we take into account ONE triangle. Considering that the area of BOTH triangles is \(\frac{1}{5}\) of 360. We must divide 180 not by 5 BUT by 2.5 (considering ONE triangle). \(\frac{180}{2.5} = 72\) Conclusion, \(a+b+c+d = 72*4 = 288\)
_________________
Get the 2 FREE GREPrepclub Tests




Re: In the figure above, the diameter of the circle is 20 and th
[#permalink]
03 Feb 2019, 01:22





