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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, 01:44
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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 511 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, 02:08
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.

Regards,
Gladi
Attachments

IMG_20180725_153135.jpg
IMG_20180725_153135.jpg [ 3.96 MiB | Viewed 440 times ]

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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, 00:05
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 294 times ]


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PKN

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Originally posted by PKN on 25 Jul 2018, 02:33.
Last edited by PKN on 26 Jul 2018, 00:05, 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 25 Jul 2018, 23:39

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] 25 Jul 2018, 23:39
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