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

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Joined: 02 Sep 2009
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In the circle above, if OA and BC are parallel, and radius OA of the  [#permalink]

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25 Jul 2018, 01:44
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Difficulty:

45% (medium)

Question Stats:

65% (02:18) correct 35% (01:36) wrong based on 37 sessions

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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 [ 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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25 Jul 2018, 02:08
Bunuel wrote:

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

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,
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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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Updated on: 26 Jul 2018, 00:05
Bunuel wrote:

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 [ 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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Joined: 04 Jan 2015
Posts: 2309
Re: In the circle above, if OA and BC are parallel, and radius OA of the  [#permalink]

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25 Jul 2018, 23:39

Solution

Given:

• OA and BC are parallel

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= $$120^0$$

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.

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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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