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The total area of the four equal circles in the figure above is 36π, a

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The total area of the four equal circles in the figure above is 36π, a  [#permalink]

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New post 10 Jan 2019, 03:19
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Question Stats:

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The total area of the four equal circles in the figure above is 36π, and the circles are all tangent to one another. What is the diameter of the small circle?


A. \(6\sqrt{2}\)

B. \(6+\sqrt{2}\)

C. \(3\sqrt{2}-3\)

D. \(6\sqrt{2}-6\)

E. \(6\sqrt{2}+6\)


Attachment:
2019-01-10_1416.png
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Re: The total area of the four equal circles in the figure above is 36π, a  [#permalink]

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New post 10 Jan 2019, 04:52
Bunuel wrote:
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The total area of the four equal circles in the figure above is 36π, and the circles are all tangent to one another. What is the diameter of the small circle?


A. \(6\sqrt{2}\)

B. \(6+\sqrt{2}\)

C. \(3\sqrt{2}-3\)

D. \(6\sqrt{2}-6\)

E. \(6\sqrt{2}+6\)

Attachment:
2019-01-10_1416.png



Area of four equal circles = 36π

i.e. Area of each bigger circle = 9π = πr^2

i.e. r = 3

Make a triangle by joining centres of circles A, B and C

the Right triangle formed has legs = 2*r = 6 units long

i.e. Hypotenuse of triangle 6√2 = 2r + Diameter of smaller circle

Diameter of smaller circle = 6√2 - 6

Answer: Option D
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Re: The total area of the four equal circles in the figure above is 36π, a  [#permalink]

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New post 10 Jan 2019, 07:35
Bunuel wrote:
Image
The total area of the four equal circles in the figure above is 36π, and the circles are all tangent to one another. What is the diameter of the small circle?


A. \(6\sqrt{2}\)

B. \(6+\sqrt{2}\)

C. \(3\sqrt{2}-3\)

D. \(6\sqrt{2}-6\)

E. \(6\sqrt{2}+6\)


Attachment:
The attachment 2019-01-10_1416.png is no longer available


Posted from my mobile device

Solution attached.
IMO D
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Re: The total area of the four equal circles in the figure above is 36π, a  [#permalink]

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New post 10 Jan 2019, 22:48
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Bunuel wrote:
Image
The total area of the four equal circles in the figure above is 36π, and the circles are all tangent to one another. What is the diameter of the small circle?


A. \(6\sqrt{2}\)

B. \(6+\sqrt{2}\)

C. \(3\sqrt{2}-3\)

D. \(6\sqrt{2}-6\)

E. \(6\sqrt{2}+6\)


Attachment:
2019-01-10_1416.png


The combined area of four equal circles is 36\(\pi\)

Each circle has area 9\(\pi\)

So, radius of each of the equal circles is 3 = r

Now, draw a square connecting the centers of the four circles.

Side of the square = 2r = 6

Let the diameter of the inner circle be x

Then, diagonal of the Square = 6\(\sqrt{2}\) = 2r +x

6\(\sqrt{2}\) = 6 + x

x = 6\(\sqrt{2}\) - 6

Choice D
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Re: The total area of the four equal circles in the figure above is 36π, a   [#permalink] 10 Jan 2019, 22:48
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