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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In the triangle OCA, CAO is 60 degrees. Hence , OCA is an equilateral traingle(60-60-60). (OC=OA=radius=CA)
Perimeter=OC+OA+AC=3*radius.
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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Regards,

PKN

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