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Geometry Prob

by sandipgumtya » Sun Jul 12, 2015 4:53 am
"ABC and CDE are two identical semi-circles of radius 2 units. B and D are the mid points of the arc ABC and CDE respectively. What is the area of the shaded region? "
(A)4pi - 1
(B)3pi - 1
(C)2pi - 4
(D)(3pi - 1)/2
(E)2pi-2
Pl find attached image.

I am getting C as ans.Pl confirm if I am right.
Are these type of qus likely to show up in GMAT?
Pl help.
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by Jim@StratusPrep » Sun Jul 12, 2015 5:45 am
Tough question, I could definitely see it as a question. See the figure below. You have 2 sectors each with an area of π and 2 triangles each with an area of 2.

2Ï€ - 4

Image


Let me know if you have questions.
[/img]
Last edited by Jim@StratusPrep on Sun Jul 12, 2015 8:04 am, edited 1 time in total.
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by theCEO » Sun Jul 12, 2015 8:03 am
Jim@StratusPrep wrote:Tough question, I could definitely see it as a question. See the figure below. You have 2 sectors each with an area of π and 2 triangles each with an area of 1.

2Ï€ - 2

Image


Let me know if you have questions.
[/img]
I think the area of the triangle should be 2 and not 1 as you wrote.

I am also getting 2Ï€ - 4 for the answer.
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by Jim@StratusPrep » Sun Jul 12, 2015 8:05 am
Ha, yup. Made the change. Thanks.
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by Brent@GMATPrepNow » Sun Jul 12, 2015 8:40 am
sandipgumtya wrote:"ABC and CDE are two identical semi-circles of radius 2 units. B and D are the mid points of the arc ABC and CDE respectively. What is the area of the shaded region? "
(A)4pi - 1
(B)3pi - 1
(C)2pi - 4
(D)(3pi - 1)/2
(E)2pi-2
Let's apply some ESTIMATION.
We can do so because, unless specified otherwise, the diagrams in Problem Solving questions are drawn to scale.
If we combine the two semi-circles, we get a complete circle with radius 2.
Area of circle = (pi)(r²) = (pi)(2²) = 4(pi) ≈ 4(3) ≈ 12
So, the total area of the two semi-circles is APPROXIMATELY 12

Now, let's ESTIMATE the portion of the semi-circles that's shaded.
Hmmm, looks like ABOUT 1/5 (or perhaps 1/6) is shaded.
1/5 of 12 = 12/5 = 2 2/5 = 2.SOMETHING

Let's check the answer choices to see if any of them are APPROXIMATELY 2.SOMETHING

(A)4pi - 1 ≈ 4(3) - 1 ≈ 11 not even close to 2.SOMETHING
(B)3pi - 1 ≈ 3(3) - 1 ≈ 8 not even close to 2.SOMETHING
(C)2pi - 4 ≈ 2(3) - 4 ≈ 2 looks good. KEEP
(D)(3pi - 1)/2 ≈ [3(3) - 1]/2 ≈ 4 not even close to 2.SOMETHING
(E)2pi-2 ≈ 2(3)-2 ≈ 4 not even close to 2.SOMETHING

Answer: C

Cheers,
Brent
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by sandipgumtya » Sun Jul 12, 2015 8:47 am
Brent@GMATPrepNow wrote:
sandipgumtya wrote:"ABC and CDE are two identical semi-circles of radius 2 units. B and D are the mid points of the arc ABC and CDE respectively. What is the area of the shaded region? "
(A)4pi - 1
(B)3pi - 1
(C)2pi - 4
(D)(3pi - 1)/2
(E)2pi-2
Let's apply some ESTIMATION.
We can do so because, unless specified otherwise, the diagrams in Problem Solving questions are drawn to scale.
If we combine the two semi-circles, we get a complete circle with radius 2.
Area of circle = (pi)(r²) = (pi)(2²) = 4(pi) ≈ 4(3) ≈ 12
So, the total area of the two semi-circles is APPROXIMATELY 12

Now, let's ESTIMATE the portion of the semi-circles that's shaded.
Hmmm, looks like ABOUT 1/5 (or perhaps 1/6) is shaded.
1/5 of 12 = 12/5 = 2 2/5 = 2.SOMETHING

Let's check the answer choices to see if any of them are APPROXIMATELY 2.SOMETHING

(A)4pi - 1 ≈ 4(3) - 1 ≈ 11 not even close to 2.SOMETHING
(B)3pi - 1 ≈ 3(3) - 1 ≈ 8 not even close to 2.SOMETHING
(C)2pi - 4 ≈ 2(3) - 4 ≈ 2 looks good. KEEP
(D)(3pi - 1)/2 ≈ [3(3) - 1]/2 ≈ 4 not even close to 2.SOMETHING
(E)2pi-2 ≈ 2(3)-2 ≈ 4 not even close to 2.SOMETHING

Answer: C

Cheers,
Brent
@Brent:Like ur approach.But how r u estimating the shaded area to be 1/5 of total.pl explain.Can we expect such question in GMAT?
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by Brent@GMATPrepNow » Sun Jul 12, 2015 8:59 am
sandipgumtya wrote: @Brent:Like ur approach.But how r u estimating the shaded area to be 1/5 of total.pl explain.Can we expect such question in GMAT?
Take 1 semi-circle, and examine the half of it (the part that has the shaded portion).
Does the shaded portion take up half of that portion of the circle?
No. So, the shaded portion takes up less than 1/4 of the semi-circle.
If you're okay with this conclusion, we can answer the question.

We're saying that LESS THAN 1/4 of the figure is shaded.
1/4 of 12 = 3
So, the area of the shaded figure is LESS THAN 3

Check the answer choices (using approximation).
Only answer choice C is less than 3.

Cheers,
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by GaneshPrasad » Mon Jul 13, 2015 5:06 am
flip the lower semi circle to the other side like this:
Image

now we just need to subtract the area of the triangle from the area of the semi circle its inscribed in,

1/2(4Ï€) - 1/2(4x2)

= 2Ï€ - 4
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