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In the circle above, if OA and BC are parallel, and radius OA of the

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In the circle above, if OA and BC are parallel, and radius OA of the  [#permalink]

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New post 25 Jul 2018, 02:44
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In the circle above, if OA and BC are parallel, and radius OA of the circle is 3, what is the length of minor arc AD?


A. \(\frac{3}{2}\pi\)

B. \(2\pi\)

C. \(\frac{5}{2}\pi\)

D. \(3\pi\)

E. \(6\pi\)


Attachment:
Untitled.png
Untitled.png [ 13.32 KiB | Viewed 388 times ]

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Re: In the circle above, if OA and BC are parallel, and radius OA of the  [#permalink]

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New post 25 Jul 2018, 03:08
Bunuel wrote:
Image
In the circle above, if OA and BC are parallel, and radius OA of the circle is 3, what is the length of minor arc AD?


A. \(\frac{3}{2}\pi\)

B. \(2\pi\)

C. \(\frac{5}{2}\pi\)

D. \(3\pi\)

E. \(6\pi\)



Attachment:
The attachment Untitled.png is no longer available


Answer is C.

The angle subtended by ARC DC at point B is half of what it subtends at the centre ( given as 60 degrees). From this we can use the parallel lines to find the central angle who minor arc is asked.

Please see attached image for detailed explanation.

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Gladi
Attachments

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In the circle above, if OA and BC are parallel, and radius OA of the  [#permalink]

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New post Updated on: 26 Jul 2018, 01:05
Bunuel wrote:
Image
In the circle above, if OA and BC are parallel, and radius OA of the circle is 3, what is the length of minor arc AD?


A. \(\frac{3}{2}\pi\)

B. \(2\pi\)

C. \(\frac{5}{2}\pi\)

D. \(3\pi\)

E. \(6\pi\)


Attachment:
The attachment Untitled.png is no longer available


Given
(i) OA=OB=OC=OD=radius of circle=3 unit (It is infered that OBC is an isosceles triangle)
(ii)OA||BC
(iii) \(\angle{OCD}\)=60 degree
To find: Length of Minor arc AD ? (Minor arc=Arc AXD in figure) (Our aim is to determine \(\angle{AOD}\))
Refer to the enclosed figure, we have \(\angle{OBC}=30\) (As per given data(i))
We have \(\angle{AOB}=\angle{OBC}=30\) (As per given data (ii)) (OA||BC; so alternate angles are equal)
Also we have \(\angle{BOD}=180=\angle{AOB}+\angle{AOD}\) (supplementary angles; angles on a straight line(here diameter))
So, \(\angle{AOD}=180-30=150\)

Now, Arc AXD=\(2πr*\frac{150}{360}\)=\(2π*3*\frac{5}{12}\)=\(\frac{5}{2}\pi\)

Ans. (C)
Attachments

Arc.JPG
Arc.JPG [ 17.23 KiB | Viewed 177 times ]


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Originally posted by PKN on 25 Jul 2018, 03:33.
Last edited by PKN on 26 Jul 2018, 01:05, edited 2 times in total.
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Re: In the circle above, if OA and BC are parallel, and radius OA of the  [#permalink]

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New post 26 Jul 2018, 00:39

Solution



Given:

    • OA and BC are parallel
    • Radius = 3

To find:
    • The length of minor arc AD.

Approach and Working:

    • The made by the arc CD made at the centre is 600.
    • Hence, angle made by arc CD at circumference is 300.
    • Thus, ∠OBC=\(30^0\)

In ∆ OBC,
    • OB=BC
    • Hence, ∠OBC= =∠OCB= \(30^0\)
    • Thus, ∠COB= \(120^0\)

Image

By alternate angles:
    • ∠OBC= ∠BOA=\(30^0\)
    • Thus, ∠AOD= \(150^0\)
    • Hence, length of the arc AD= \(\frac{150}{360}*2*π *3= \frac{5}{2} π\)

Hence, the correct answer is option C.

Answer: C
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Re: In the circle above, if OA and BC are parallel, and radius OA of the &nbs [#permalink] 26 Jul 2018, 00:39
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