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655-705 (Hard)|   Geometry|      
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Bunuel
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In the circle above, if OA and BC are parallel, and radius OA of the 25 Jul 2018, 02:44
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C
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55% (hard)

Question Stats:

64% (03:09) correct 36% (02:28) wrong based on 70 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. \(6\pi\)


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C
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In the circle above, if OA and BC are parallel, and radius OA of the 25 Jul 2018, 03:08
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Bunuel

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. \(6\pi\)



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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
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In the circle above, if OA and BC are parallel, and radius OA of the Updated on: 26 Jul 2018, 01:05
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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Bunuel

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. \(6\pi\)


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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)
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In the circle above, if OA and BC are parallel, and radius OA of the 26 Jul 2018, 00:39
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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= \(120^0\)



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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In the circle above, if OA and BC are parallel, and radius OA of the 15 Dec 2022, 06:39
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Gladiator59
Bunuel

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. \(6\pi\)



Show Spoiler
Attachment:
Untitled.png

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
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Gladiator59
The question doesn't mention that the line DB is the diameter of that circle. If line DB isn't the diameter, please explain the rationale behind angle B = 30
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