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01 May 2021, 08:15
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E
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Difficulty:
85%
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Question Stats:
41% (02:17) correct
59%
(02:20)
wrong
based on 46
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If a circle is circumscribed around a regular pentagon, as shown above, what is the value of x?
(A) 72
(B) 108
(C) 135
(D) 140
(E) 144
Attachment:
1.png [ 19.8 KiB | Viewed 3500 times ]
01 May 2021, 11:50
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We know angle D = angle E = 120 (regular pentagon)
Let the point at which x intersects the circle be O.
Construct line XY parallel to ED.
Consider pentagon XYDOE.
Angle EXY = 60 (since angle AXY=120 by construction)
Similarly angle XYD = 60
Since sum of internal angles of a pentagon = 540
60+60+120+120+x=540
or, x = 140
Answer = D
Let the point at which x intersects the circle be O.
Construct line XY parallel to ED.
Consider pentagon XYDOE.
Angle EXY = 60 (since angle AXY=120 by construction)
Similarly angle XYD = 60
Since sum of internal angles of a pentagon = 540
60+60+120+120+x=540
or, x = 140
Answer = D
Rahulprabhu
Joined: 01 Dec 2019
Last visit: 03 Apr 2022
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Location: India
Schools: Anderson '24 Columbia CBS '24 Darden '24 Fuqua '24 Haas '24 Harvard '24 Kellogg '24 LBS '24 Marshall '24 Ross '24 Sloan '24 Stanford '24 Tepper '24 Wharton '24 Yale '24
GMAT 1: 710 Q50 V35
Schools: Anderson '24 Columbia CBS '24 Darden '24 Fuqua '24 Haas '24 Harvard '24 Kellogg '24 LBS '24 Marshall '24 Ross '24 Sloan '24 Stanford '24 Tepper '24 Wharton '24 Yale '24
GMAT 1: 710 Q50 V35
Posts: 3
01 May 2021, 12:53
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Since a regular Pentagon has all sides equal and all the vertices of the petagon lie on the circle.The vertices are equidistant from the center of the circle.
Let the center of the circle be O.
Now joining the vertices of the Pentagon to the center O, we get 5 equal triangles.
Since the circle has 360 deg.
Each angle, i.e AOB=BOC=COD=DOE=EOA=360/5= 72 deg.
Since the angle DOE and EXD originate from the same chord.
X= DOE*2=72*2= 144
IMO E
ankitanand12 I guess you've not considered the two small angles in the triangle DEX. I may be wrong.
Posted from my mobile device
Let the center of the circle be O.
Now joining the vertices of the Pentagon to the center O, we get 5 equal triangles.
Since the circle has 360 deg.
Each angle, i.e AOB=BOC=COD=DOE=EOA=360/5= 72 deg.
Since the angle DOE and EXD originate from the same chord.
X= DOE*2=72*2= 144
IMO E
ankitanand12 I guess you've not considered the two small angles in the triangle DEX. I may be wrong.
Posted from my mobile device
