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Question 2 :In a circle with centre O and radius 1 cm, an arc AB makes an angle 60 degrees at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two points on OA and OB, respectively, such that OC = OD and the area of triangle OCD is half that of R, then the length of OC, in cm, is

  1. \\frac{π}{4})\\frac{1}{2})
  2. \\frac{π}{6})\\frac{1}{2})
  3. \\frac{π}{4√3})\\frac{1}{2})
  4. \\frac{π}{3√3})\\frac{1}{2})

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Explanatory Answer

Method of solving this CAT Question from Geometry
Geometry - Finding the length of OC

Given ∠AOB = 60°
Area of Sector AOB = \\frac{60}{360}) × π × 12 = \\frac{π}{6}) ---(1)
Given OC = OD => ∠OCD = ∠ODC = 60°
△OCD is an Equilateral Triangle with side = a
Area(△OCD) = \\frac{√3}{4}) × a × a ---(2)
Its given that Area(OCD) = \\frac{1}{2}) × Area(OAB)
a2\\frac{√3}{4}) = \\frac{π}{6×2})
a = (\\frac{π}{3√3}))\\frac{1}{2})

The question is "In a circle with centre O and radius 1 cm, an arc AB makes an angle 60 degrees at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two points on OA and OB, respectively, such that OC = OD and the area of triangle OCD is half that of R, then the length of OC, in cm, is"

Hence, the answer is \\frac{π}{3√3})\\frac{1}{2}) cm

Choice D is the correct answer.

 

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