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Area of shaded region of circle

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Area of shaded region of circle

by gmattesttaker2 » Sun Feb 16, 2014 12:30 am
Hello,

Can you please assist with this:

In the figure above, if the area of the circle with center N has area pi, and
MNP is a diameter of the circle, what is the area of the shaded region?

(1) MON and NQP are semi-circles.
(2) The shaded area equals the unshaded area.

OA: D

I started as follows:

Area of circle = pi
=> pi r^2 = pi
=> r^2 = 1
=> r = 1

1) Area of semi circle MON = pi (1/2)^2 = pi(1/4)

However, I was not sure how to proceed from here.
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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Sun Feb 16, 2014 9:09 am
gmattesttaker2 wrote: Image
In the figure above, if the area of the circle with center N has area pi, and
MNP is a diameter of the circle, what is the area of the shaded region?

(1) MON and NQP are semi-circles.
(2) The shaded area equals the unshaded area.
Target question: What is the area of the shaded region?

Given: N is the center of the circle. The circle has area pi. MNP is a diameter of the circle.

Area = (pi)r²
So, if the area of this circle is pi, we get: pi = (pi)r², which means r = 1
So, let's add this to our diagram.
Image

NOTE: At this point, we might conclude that we already have enough information to determine the area of the shaded region, but we really don't know anything about the shapes MON and NQP. They COULD be semicircles or the COULD be something else, like differently-shaped ellipses.

Statement 1: MON and NQP are semi-circles.
At this point, we COULD determine the areas of MON and NQP, but we don't have to.
Just recognize that MON and NQP (the blue areas below) must have the same areas since they're both semicircles with the same radii.
Image

So, we could just take the shaded region MON and use it to fill the gap created by NQP to get the following:
Image

At this point, we can see that the shaded region takes up HALF the space of the entire circle.
Since we already know the ENTIRE circle has area pi, the shaded portion must have area pi/2
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The shaded area equals the unshaded area.
In other words, the shaded region takes up HALF the space of the entire circle.
Since we already know the ENTIRE circle has area pi, the shaded portion must have area pi/2
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by gmattesttaker2 » Sun Feb 16, 2014 6:21 pm
Brent@GMATPrepNow wrote:
gmattesttaker2 wrote: Image
In the figure above, if the area of the circle with center N has area pi, and
MNP is a diameter of the circle, what is the area of the shaded region?

(1) MON and NQP are semi-circles.
(2) The shaded area equals the unshaded area.
Target question: What is the area of the shaded region?

Given: N is the center of the circle. The circle has area pi. MNP is a diameter of the circle.

Area = (pi)r²
So, if the area of this circle is pi, we get: pi = (pi)r², which means r = 1
So, let's add this to our diagram.
Image

NOTE: At this point, we might conclude that we already have enough information to determine the area of the shaded region, but we really don't know anything about the shapes MON and NQP. They COULD be semicircles or the COULD be something else, like differently-shaped ellipses.

Statement 1: MON and NQP are semi-circles.
At this point, we COULD determine the areas of MON and NQP, but we don't have to.
Just recognize that MON and NQP (the blue areas below) must have the same areas since they're both semicircles with the same radii.
Image

So, we could just take the shaded region MON and use it to fill the gap created by NQP to get the following:
Image

At this point, we can see that the shaded region takes up HALF the space of the entire circle.
Since we already know the ENTIRE circle has area pi, the shaded portion must have area pi/2
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The shaded area equals the unshaded area.
In other words, the shaded region takes up HALF the space of the entire circle.
Since we already know the ENTIRE circle has area pi, the shaded portion must have area pi/2
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent
Hello Brent,

Thank you very much for your excellent explanation (and for the superb visuals as well.)

Best Regards,
Sri
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