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In the above diagram, the circle inscribes

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In the above diagram, the circle inscribes [#permalink] New post 08 Jun 2018, 05:30
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In the above diagram, the circle inscribes the larger equilateral, and it circumscribes the smaller equilateral triangle. If the area of the smaller triangle is √3, what is the area of the larger triangle?

A) 9π - 16√3
B) 4√3
C) 8√3
D) 16√3
E) 16π - 2√3
[Reveal] Spoiler: OA

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Re: In the above diagram, the circle inscribes [#permalink] New post 10 Jun 2018, 06:33
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GreenlightTestPrep wrote:
Image
In the above diagram, the circle inscribes the larger equilateral, and it circumscribes the smaller equilateral triangle. If the area of the smaller triangle is √3, what is the area of the larger triangle?

A) 9π - 16√3
B) 4√3
C) 8√3
D) 16√3
E) 16π - 2√3


We're told that the area of the smaller triangle is √3
USEFUL FORMULA: Area of an equilateral triangle = (√3)(side²)/4
So, we can write: (√3)(side²)/4 = √3
Divide both sides by √3 to get: (side²)/4 = 1
Multiply both sides by 4 to get: side² = 4
Solve: side = 2
So, each side of the smaller equilateral triangle has length 2

Using this information, we can create a 30-60-90 triangle (in blue)
Image

We can now compare this blue 30-60-90 triangle with the BASE 30-60-90 triangle
Image
By the property of similar triangles, we know that the ratios of corresponding sides will be equal.
That is: 1/√3 = r/2
Cross multiply to get: (√3)(r) = 2
Solve: r = 2/√3
So, the RADIUS of the circle = 2/√3
We'll add this information to our diragram
Image

At this point, we can focus our attention on the GREEN 30-60-90 triangle
Image
Since we already know that the RADIUS of the circle = 2/√3, we can apply the property of similar triangles again.
The ratios of corresponding sides will be equal.
So, we get: (2/√3)/1 = (x)/√3
Cross multiply to get: (2/√3)(√3) = (x)(1)
Simplify: x = 2
Since x = HALF the length of one side of the larger triangle, we know that the ENTIRE length = 4

What is the area of the larger triangle?
We'll re-use our formula that says: area of an equilateral triangle = (√3)(side²)/4
Area = (√3)(4²)/4
= (√3)(16)/4
= 4√3

Answer: B

Cheers
Brent
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Re: In the above diagram, the circle inscribes [#permalink] New post 10 Jun 2018, 08:09
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This was really tough.

Thank you.

Regards
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1 KUDOS received
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Re: In the above diagram, the circle inscribes [#permalink] New post 10 Jun 2018, 09:13
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Carcass wrote:
This was really tough.

Thank you.

Regards


Agreed - very tough!!
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Re: In the above diagram, the circle inscribes   [#permalink] 10 Jun 2018, 09:13
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