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If an equilateral triangle is inscribed in a larger circle and is

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If an equilateral triangle is inscribed in a larger circle and is  [#permalink]

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New post 11 Jun 2019, 11:08
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  35% (medium)

Question Stats:

78% (02:48) correct 22% (02:46) wrong based on 9 sessions

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Re: If an equilateral triangle is inscribed in a larger circle and is  [#permalink]

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New post 11 Jun 2019, 11:46
indu1954 wrote:
Need explanantion


Let ABC are the vertices with vertices A on top.

Property of equilateral triangle is centroid = center of incircle = Center of circumcircle

Centroid divides Median in ratio 2:1
So, if radius of incircle = m
Radius of circumcircle = 2m
—> OA = OB = OC = 2m

Required shaded area = area of sector OBC - area of triangle OBC.

OBC is isosceles triangle with 120-30-30 angles (sides ratio 3^1/2:1:1) . If OB = OC = 2m
—> BC = 2*3^1/2*m

Area of sector = 120/360*pi*(2m)^2
= 4/3*pi*m^2

Area of triangle = 1/2*m*2*3^1/2*m
= sqrt(3)*m^2

Required Area = (4/3*pi - sqrt(3))*m^2
= (4*pi - 3*sqrt(3))*m^2/3

IMO Option A


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Re: If an equilateral triangle is inscribed in a larger circle and is  [#permalink]

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New post 11 Jun 2019, 11:57
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If we find the area of the big circle, and subtract the area of the triangle, we'll get the total area of the three equal sections outside of the triangle and inside the big circle. We only want the area of one of those sections, so if we do that subtraction, we'll need to divide by 3. So we want to find:

(Area of big circle - Area of triangle)/3

Now if you look at the small triangle at the top right of the diagram, the one with the dotted lines as two of its edges, notice that the line of length m in that triangle must meet the big equilateral triangle at 90 degrees, because the edge of the equilateral is tangent to the circle (and tangent lines are always perpendicular to radii at the tangent point). So that dotted-line triangle is a right triangle, and since we're bisecting the 60 degree angle at the top of the diagram, it must be a 30-60-90 triangle, and its sides are in a 1 to √3 to 2 ratio. Its shortest side is m, so its longest side, the vertical dotted line, is 2m. But that's the radius of the big circle, so the area of the big circle is π (2m)^2 = 4πm^2

Notice also that the third side of that small triangle, the downwards sloping side on the right, has length √3m. But that's half the length of a side of the equilateral triangle, so each side of the equilateral has length 2√3m. We need the area of that triangle, so we need to find a height (unless you know an equilateral triangle formula). If we take the horizontal side at the bottom as our base, then the vertical height is just made up of one radius of the small circle, of length m, plus the radius of the big circle which we found above was 2m. So the height of the triangle is 3m. Since our base is just the length of a side in the equilateral triangle, our base is 2√3m, and the area of the equilateral triangle is bh/2 = (2√3m)(3m)/2 = 3√3m^2.

Finally,

(Area of big circle - Area of triangle)/3 = (4πm^2 - 3√3m^2)/3 = (4π - 3√3)m^2/3
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Re: If an equilateral triangle is inscribed in a larger circle and is  [#permalink]

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New post 11 Jun 2019, 12:40
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indu1954 wrote:
Need explanantion


indu1954
please see attached solution
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Re: If an equilateral triangle is inscribed in a larger circle and is  [#permalink]

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New post 11 Jun 2019, 16:21
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Hi indu1954,

This question can be approached with a mix of TESTing VALUES and Geometry rules. To start, we're going to focus on the Equilateral triangle:

1) We can split that triangle into 3 equal pieces (break the center of the circle into three 120-degree angles) and you'll have 3 equal triangles with angles of 30/30/120. That central 120-degree angle that 'points down' will create exactly 1/3 of the circle (and that piece contains the shaded area that we're interested in.)

2) Let's TEST M = 1. When the radius of the little circle is 1, then the 'height' of the 30/30/120 triangle will equal 1 and the 'base' will be 2√3. You can calculate that by breaking the 30/30/120 triangle into two 30/60/90 triangles.

The area of that 30/30/120 triangle is --> (1/2)(base)(height) = (1/2)(2√3)(1) = √3

You also now have the radius of the big circle (since the radius of the big circle equals the hypotenuse of the 30/60/90 triangles). That larger radius is 2.

3) We can now find the area of the BIG circle --> Area = π(R^2) = π(2^2) = 4π. One third of that circle is 4π/3.

4) To find the area of the shaded region, we subtract the area of the triangle from the area of 1/3 of the big circle:

4π/3 - √3 =
4π/3 - 3√3/3 =
(4π - 3√3)/3

Since M=1, it won't be difficult to find the one answer that matches...



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Re: If an equilateral triangle is inscribed in a larger circle and is  [#permalink]

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New post 12 Jun 2019, 00:10
indu1954 wrote:
Attachment:
Equilateral Triangle.jpg


Responding to a pm:

My first thought is this: the arc of the area of the shaded region subtends 60 degree angle so the central angle it subtends must be 120 degrees.

Shaded Area = (1/3)*Area of big circle - (1/3)*Area of triangle

Side of equilateral triangle = 2*sqrt(3)*Radius of incircle = 2*sqrt(3)*m
Area of triangle = \(\frac{\sqrt{3}}{4} * side^2 = \frac{\sqrt{3}}{4} * [2*\sqrt{3}*m]^2 = 3*\sqrt{3}*m^2\)

Side of equilateral triangle = sqrt(3)*Radius of circumcircle
2*sqrt(3)*m = sqrt(3)*Radius of circumcircle
Radius of circumcircle = 2m

Area of big circle = \(\pi*radius^2 = \pi*(2m)^2\)

Shaded Region = \((1/3)*[\pi*4*m^2 - 3*\sqrt{3}*m^2]\)

Answer (A)

Check these posts for discussion on the given relations:
https://www.veritasprep.com/blog/2013/0 ... relations/
https://www.veritasprep.com/blog/2013/0 ... other-way/
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Re: If an equilateral triangle is inscribed in a larger circle and is   [#permalink] 12 Jun 2019, 00:10
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