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The number of inches in the perimeter of an equilateral triangle equal

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The number of inches in the perimeter of an equilateral triangle equal  [#permalink]

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New post 19 Mar 2019, 02:38
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The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?


(A) \(\frac{3\sqrt{2}}{\pi}\)

(B) \(\frac{3\sqrt{3}}{\pi}\)

(C) \(\sqrt{3}\)

(D) \(\frac{6}{\pi}\)

(E) \(\sqrt{3}*\pi\)

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Re: The number of inches in the perimeter of an equilateral triangle equal  [#permalink]

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New post 19 Mar 2019, 03:10
Bunuel wrote:
The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?


(A) \(\frac{3\sqrt{2}}{\pi}\)

(B) \(\frac{3\sqrt{3}}{\pi}\)

(C) \(\sqrt{3}\)

(D) \(\frac{6}{\pi}\)

(E) \(\sqrt{3}*\pi\)


The question states that the numerical value of perimeter of the equilateral triangle is equal to the value of the area of the circumscribed circle.

Let side of the triangle be = x

Hence perimeter = 3x

Height of the triangle =\(\sqrt{3}x/2\)

In case of equilateral triangle the height and median are the same lines. And the centroid is also the circumcentre. Hence the centre of the circle will be same as centroid.

We know the median of the triangle is divided by the centroid in the ration 2:1.

So the radius of the circumcircle = \(2/3*\sqrt{3}x/2 = x/\sqrt{3}\)...i)

\(Area = \pi(x/\sqrt{3}) ^2\)

From question Area= Perimeter

\(\pi(x/\sqrt{3}) ^2 = 3x\)
\(=> x = 9/\pi\)

From ...i)

Radius of the circumcircle =\(9/(\pi * \sqrt{3})\)
=\(3\sqrt{3}/\pi\)

Answer (B)
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Re: The number of inches in the perimeter of an equilateral triangle equal  [#permalink]

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New post 19 Mar 2019, 03:26
Bunuel wrote:
The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?


(A) \(\frac{3\sqrt{2}}{\pi}\)

(B) \(\frac{3\sqrt{3}}{\pi}\)

(C) \(\sqrt{3}\)

(D) \(\frac{6}{\pi}\)

(E) \(\sqrt{3}*\pi\)


As discussed here: https://www.veritasprep.com/blog/2013/0 ... relations/
If we have an equilateral triangle inscribed in a circle,
Side of the triangle = sqrt(3) * Radius of the circle

Given: Perimeter = 3*Side = Area of Circumscribed Circle = pi*Radius^2
3 * sqrt(3)*Radius = pi*Radius^2

Radius = 3*sqrt(3)/pi

Answer (B)
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Re: The number of inches in the perimeter of an equilateral triangle equal  [#permalink]

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New post 19 Mar 2019, 10:49
Bunuel wrote:
The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?


(A) \(\frac{3\sqrt{2}}{\pi}\)

(B) \(\frac{3\sqrt{3}}{\pi}\)

(C) \(\sqrt{3}\)

(D) \(\frac{6}{\pi}\)

(E) \(\sqrt{3}*\pi\)



Side of the triangle = sqrt(3) * Radius of the circle

so
radius = s /√3

given
3s=pi * r^2
r^2 = 3s/pi
s^2/3 = 3s/pi
so s= 9pi
or say
radius = 9pi/√3 = \(\frac{3\sqrt{3}}{\pi}\)
IMO B
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Re: The number of inches in the perimeter of an equilateral triangle equal   [#permalink] 19 Mar 2019, 10:49
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