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If O is the center and AB parallel to CD, then x=

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Math Expert
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Joined: 02 Sep 2009
Posts: 62353
If O is the center and AB parallel to CD, then x=  [#permalink]

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New post 18 Feb 2020, 05:16
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

64% (02:38) correct 36% (03:04) wrong based on 22 sessions

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Director
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Joined: 16 Jan 2019
Posts: 593
Location: India
Concentration: General Management
WE: Sales (Other)
If O is the center and AB parallel to CD, then x=  [#permalink]

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New post 18 Feb 2020, 05:34
AB and ED are intersecting lines and so

\(\angle AOE=\angle BOD\) and
\(\angle EOB=\angle AOD\)

We have \(\angle AOE+\angle BOD+\angle EOB+\angle AOD=360\)

Which means \(20+20+2*\angle AOD=360\)

\(\angle AOD=160\)

Since OA and OD are radii, \(\angle OAD=\angle ODA\)

In iscoceles triangle OAD, \(160+2*\angle OAD=180\)

\(\angle OAD=10\)

If AB and CD are parallel \(\angle OAD=\angle ADC\)

Since ED is a diameter, triangle DCE is a right triangle

So \(10+90+x=180\)

\(x=80\)

Answer is (E)

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Director
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Re: If O is the center and AB parallel to CD, then x=  [#permalink]

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New post 18 Feb 2020, 06:25
ED — a diameter of the circle
—> ECD — right-angled triangle: angle C = 90 degrees

AB || CD —> angle CDE = 20 degrees

Now, we got XCD right-angled triangle
—> angle D is less than 20 degrees —> x is greater than 70 degrees (x> 70)

Only the answer choice E satisfies.

The answer is E

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GMATWhiz Representative
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Re: If O is the center and AB parallel to CD, then x=  [#permalink]

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New post 18 Feb 2020, 22:03
Bunuel wrote:
Image
If O is the center and AB parallel to CD, then x=

(A) 40
(B) 50
(C) 60
(D) 70
(E) 80


Attachment:
Untitled.png


Solution:


    • As shown in the below diagram, x° is an interior angle of triangle CFD.
Image
      o x° + ∠FCD + ∠FDC = 180°
      o Or, \(x° + y° + z° = 180° ……….Eq.(i)\)
    • Now, Arc BD subtends ∠BOD at centre and ∠BAD at a circumference point.
      o \(∠BOD = 2*∠BAD = 20° ⟹ ∠BAD = 10° …….Eq.(ii)\)
    • Since line AB is parallel to CD,
      o ∠ADC or y = ∠BAD (Alternate interior angles)
      o Substituting the value ∠BAD from Eq.(ii into the above equation, we get
         \(y = 10° …………Eq.(iii)\)
    • Now, triangle CDE is a tringle circumscribed in half circle and has one of its side as diameter of the circle.
      o Therefore, triangle CDE is a right-angled triangle, right angled at C.
      o Or, ∠ECD or \(z = 90° …….Eq.(iv)\)
    • Substituting the value of y and z from Eq.(iii) and (iv) into Eq.(i), we get
      o \(x° + 10° + 90° = 180°\)
      o Thus, \(x = 80\)
Thus, the correct answer is Option E.
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Re: If O is the center and AB parallel to CD, then x=   [#permalink] 18 Feb 2020, 22:03
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