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# In the figure above, the center of the circle is O, and the measure...

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In the figure above, the center of the circle is O, and the measure...  [#permalink]

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14 Jul 2018, 07:25
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Difficulty:

35% (medium)

Question Stats:

68% (01:31) correct 32% (01:34) wrong based on 37 sessions

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2018-07-14_20-43-15.jpg [ 9.85 KiB | Viewed 423 times ]

In the figure above, the center of the circle is O, and the measure of angle CAO is 60 degrees. What is the perimeter of triangle OAC?

1. The length of arc CDA is $$16\pi$$ greater than the length of arc ABC.

2. The area of triangle OAC is $$36\sqrt{3}$$

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Manish

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Re: In the figure above, the center of the circle is O, and the measure...  [#permalink]

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14 Jul 2018, 07:44
CAMANISHPARMAR wrote:
Attachment:
2018-07-14_20-43-15.jpg

In the figure above, the center of the circle is O, and the measure of angle CAO is 60 degrees. What is the perimeter of triangle OAC?

1. The length of arc CDA is $$16\pi$$ greater than the length of arc ABC.

2. The area of triangle OAC is $$36\sqrt{3}$$

$$\angle{OAC}=\angle{OCA} = 60$$, therefore angle COA is 180-60-60=60
therefore Triangle OAC is equilateral and each side is equal to the radius and therefore the perimeter is 3*radius

QUESTION :- what is the radius of circle?

1. The length of arc CDA is $$16\pi$$ greater than the length of arc ABC.
arc ABC has an angle 360-60=300 at O, since angle COA is 60
and arc is subtended by 60....
so (300-60) degree =240= 16 pi.......
so 360 degree = circumference = 16*360/240=24 *pi =2*pi*radius..
sufficient

2. The area of triangle OAC is $$36\sqrt{3}$$[
since it is equilateral triangle, area depends only on side which is equal to radius of circle
since area is given, radius can be found
suff

D
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In the figure above, the center of the circle is O, and the measure...  [#permalink]

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14 Jul 2018, 08:08
CAMANISHPARMAR wrote:
Attachment:
2018-07-14_20-43-15.jpg

In the figure above, the center of the circle is O, and the measure of angle CAO is 60 degrees. What is the perimeter of triangle OAC?

1. The length of arc CDA is $$16\pi$$ greater than the length of arc ABC.

2. The area of triangle OAC is $$36\sqrt{3}$$

In the triangle OCA, CAO is 60 degrees. Hence , OCA is an equilateral traingle(60-60-60). (OC=OA=radius=CA)
We need to check sufficiency of determining radius of circle.

Statement1:-The length of arc CDA is $$16\pi$$ greater than the length of arc ABC.

Or. 2$$\pi$$r($$\frac{5}{6}$$)=2$$\pi$$r($$\frac{1}{6}$$)+16$$\pi$$ (Arc CDA is 5 parts out of 6 with each part is equal to 60 degree & Arc CBA is one part out of 6 parts)
Or, $$r(\frac{5}{6}-\frac{1}{6})=8$$
Or, r=12 unit
So, perimeter can be determined.
Hence sufficient.

Statement-2:- The area of triangle OAC is $$36\sqrt{3}$$

Since OAC is an equilateral triangle, hence side of equilateral triangle can be determined from a given area.

$$Area=(\sqrt{3}/4)*OC^2$$

Any side of triangle OAC= radius of circle.
Hence Perimeter can be determined.
Exact computation of perimeter is not required.
Hence, sufficient.

Ans. (D)
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In the figure above, the center of the circle is O, and the measure... &nbs [#permalink] 14 Jul 2018, 08:08
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