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Arcs AC, AB and BC are centered at B, C and A respectively, as shown a

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Arcs AC, AB and BC are centered at B, C and A respectively, as shown a [#permalink]

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New post 11 Jun 2015, 11:15
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Attachment:
T7908.png
T7908.png [ 10.93 KiB | Viewed 1387 times ]

Arcs AC, AB and BC are centered at B, C and A respectively, as shown above. If the area of equilateral triangle ABC is √3, what is the area of the shaded region?

A. \(2pi-3\sqrt{3}\)
B. \(\sqrt{3} - pi\)
C. 3pi
D. \(\sqrt{3}+2pi\)
E. 3

[Reveal] Spoiler:
Official Help:
Shaded region problems call for Ballparking and eliminating irrelevant answer choices.
Use Ballparking and avoid unnecessary calculations. The shaded area is the target value that you seek. In your ballparking calculations you use the given value - in our case it's √3, the area of the triangle. Run on each answer using elimination criteria, from the faster elimination to the slower.
Start by checking if the expression is negative - An area can't be negative.
Next check if the format is correct - in this question the shaded area is the result of "subtraction with pi", don't waste time on trying to figure out what is subtracted from what. Just note that you expect a subtraction and that one of the members of the expression should include ∏.
Next ballpark for estimated size and POE answers that are not in the ballpark.

After folding the shaded regions onto the triangle it is easier to guesstimate the area of the shaded region:
Attachment:
T7908B.png
T7908B.png [ 12.31 KiB | Viewed 1380 times ]

The shaded regions add up to approximately half the area of the triangle. If you're not sure you see that, subdivide the triangle connecting the center of the circle with vertices A, B, C as shown it this figure:
Attachment:
T7908c.png
T7908c.png [ 9.98 KiB | Viewed 1380 times ]

Therefore, your target is half of the triangle= √3/2≈0.9
[Reveal] Spoiler: OA

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Last edited by reto on 12 Jun 2015, 02:35, edited 1 time in total.

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Re: Arcs AC, AB and BC are centered at B, C and A respectively, as shown a [#permalink]

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New post 11 Jun 2015, 20:59
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reto wrote:
Attachment:
T7908.png

Arcs AC, AB and BC are centered at B, C and A respectively, as shown above. If the area of equilateral triangle ABC is √3, what is the area of the shaded region?


[Reveal] Spoiler:
Official Help:
Shaded region problems call for Ballparking and eliminating irrelevant answer choices.
Use Ballparking and avoid unnecessary calculations. The shaded area is the target value that you seek. In your ballparking calculations you use the given value - in our case it's √3, the area of the triangle. Run on each answer using elimination criteria, from the faster elimination to the slower.
Start by checking if the expression is negative - An area can't be negative.
Next check if the format is correct - in this question the shaded area is the result of "subtraction with pi", don't waste time on trying to figure out what is subtracted from what. Just note that you expect a subtraction and that one of the members of the expression should include ∏.
Next ballpark for estimated size and POE answers that are not in the ballpark.

After folding the shaded regions onto the triangle it is easier to guesstimate the area of the shaded region:
Attachment:
T7908B.png

The shaded regions add up to approximately half the area of the triangle. If you're not sure you see that, subdivide the triangle connecting the center of the circle with vertices A, B, C as shown it this figure:
Attachment:
T7908c.png

Therefore, your target is half of the triangle= √3/2≈0.9


You need to give the options.

Let's focus on arc BC and its centre at A. The circle which has A as the centre and BC as an arc has radius AB - the side of equilateral triangle.
The area of the equilateral triangle = \(\sqrt{3}/4 * a^2 = \sqrt{3}\) so a, the side will be 2.

Triangle ABC is equilateral so angle A is 60 degrees. Effectively, arc BC is subtending a 60 degree angle at the centre.
So area of sector ABC will be 60/360 * Area of circle = \((1/6) * \pi * 2^2 = 2\pi/3\)

The shaded region area = 3* (Area of sector ABC - Area of triangle ABC) = \(3*(2\pi/3 - \sqrt{3}) = 2\pi - 3\sqrt{3}\)
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Re: Arcs AC, AB and BC are centered at B, C and A respectively, as shown a [#permalink]

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New post 11 Jun 2015, 21:00
reto wrote:
Attachment:
T7908.png

Arcs AC, AB and BC are centered at B, C and A respectively, as shown above. If the area of equilateral triangle ABC is √3, what is the area of the shaded region?


[Reveal] Spoiler:
Official Help:
Shaded region problems call for Ballparking and eliminating irrelevant answer choices.
Use Ballparking and avoid unnecessary calculations. The shaded area is the target value that you seek. In your ballparking calculations you use the given value - in our case it's √3, the area of the triangle. Run on each answer using elimination criteria, from the faster elimination to the slower.
Start by checking if the expression is negative - An area can't be negative.
Next check if the format is correct - in this question the shaded area is the result of "subtraction with pi", don't waste time on trying to figure out what is subtracted from what. Just note that you expect a subtraction and that one of the members of the expression should include ∏.
Next ballpark for estimated size and POE answers that are not in the ballpark.

After folding the shaded regions onto the triangle it is easier to guesstimate the area of the shaded region:
Attachment:
T7908B.png


The shaded regions add up to approximately half the area of the triangle. If you're not sure you see that, subdivide the triangle connecting the center of the circle with vertices A, B, C as shown it this figure:
Attachment:
T7908c.png

Therefore, your target is half of the triangle= √3/2≈0.9


There are three shaded areas of equal area.

Area of Equilateral Triangle = \((\sqrt{3}/4) side^2 = \sqrt{3}\)
i.e. Side of Triangle = 2 = Radius of arc

One shaded area = Area of sector - Area of Equilateral Triangle
One shaded area = \((\frac{60}{360})(pi)R^2 - \sqrt{3}\)
One shaded area = \((\frac{60}{360})(pi)2^2 - \sqrt{3}\)
One shaded area = \((\frac{2}{3})(pi) - \sqrt{3}\)

Three Shaded Area = \(3*(\frac{2}{3})(pi) - \sqrt{3}\)
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Re: Arcs AC, AB and BC are centered at B, C and A respectively, as shown a [#permalink]

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Arcs AC, AB and BC are centered at B, C and A respectively, as shown a [#permalink]

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New post 25 Oct 2017, 11:13
reto wrote:
Attachment:
T7908.png

Arcs AC, AB and BC are centered at B, C and A respectively, as shown above. If the area of equilateral triangle ABC is √3, what is the area of the shaded region?

A. \(2pi-3\sqrt{3}\)
B. \(\sqrt{3} - pi\)
C. 3pi
D. \(\sqrt{3}+2pi\)
E. 3

1) Choose one vertex and its segment, say B

2) Area of shaded region for B:
(Area of segment) - (Area of triangle)

3) Segment area - find SIDE LENGTH of equilateral triangle, which is the RADIUS of Circle B:

Find side length
Area of equilateral triangle, given: \(\sqrt{3}\)
Formula for that area:\(\frac{s^2\sqrt{3}}{4}\)

\(\frac{s^2\sqrt{3}}{4}\)=\(\sqrt{3}\)

\(s^2\sqrt{3}=4\sqrt{3}\)

\(s^2 =\frac{4\sqrt{3}}{\sqrt{3}}\)

\(s^2 = 4\)
\(s = 2\)

So radius of Circle B = 2

3) SEGMENT AREA: Find Circle B's area. What fraction does segment area equal?

Circle area: \(\pi*r^2 = 4\pi\)

Segment area as fraction? An equilateral triangle has 3 vertices with angle measures of 60°

Fraction:\(\frac{60}{360}=\frac{1}{6}\) of circle's area
Segment area: \((\frac{1}{6})(4\pi)=\frac{2}{3}\pi\)

4) Shaded area * 3
Areas: Shaded = (Segment) - (triangle)
One shaded area\(:\frac{2}{3}\pi -\sqrt{3}\)
There are three such areas, so total shaded area is
\(3(\frac{2}{3}\pi-\sqrt{3})=2\pi-3\sqrt{3}\)

Answer A

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Arcs AC, AB and BC are centered at B, C and A respectively, as shown a   [#permalink] 25 Oct 2017, 11:13
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