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In the figure above, ABCD is a rectangle and DA and CB are radii of

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Bunuel
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In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   16 Mar 2018, 00:08
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In the figure above, ABCD is a rectangle and DA and CB are radii of the circles shown. If AB = 4, what is the perimeter of the shaded region?

(A) 2π + 4

(B) 4π + 4

(C) 4π + 8

(D) 8π + 8

(E) 8π + 16


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In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   21 Mar 2018, 00:57
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Solution:



Let us assume, the point of contact of both the circles is P.



Perimeter of Shaded region= AB + length of arc AP +length of Arc BP

    • \(AB=DC= 4\)

    • \(DC= r+ r=2r=4\), where r is the radius of the circle.

    • \(r=2\)

Length of Arc AP:

    • \(AP= (90/ 360) * 2* π* 2\)

    • \(AP= ¼ * 2* π*2\)

    • \(AP= π\)

Length of Arc BP:

    • Since arc BP also makes an angle of \(900\) at the center and made by the radius=\(2\), the length of arc BP is equal to\(π\).

      o Length of arc AP +length of Arc BP= \(π+π= 2π\)

The perimeter of Shaded region=\(4+ 2π\). Hence (A) is the correct answer.

Answer: A
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In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   16 Mar 2018, 01:14
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Bunuel wrote:

In the figure above, ABCD is a rectangle and DA and CB are radii of the circles shown. If AB = 4, what is the perimeter of the shaded region?

(A) 2π + 4

(B) 4π + 4

(C) 4π + 8

(D) 8π + 8

(E) 8π + 16


Show Spoiler
Attachment:
2018-03-16_1011_004.png


Shaded area = Rectangle - Two quarter circles

AB = 2* Radius of each circle = 4

i.e. Radius of each circle = 2

i.e. Dimensions of rectangle are 4 and 2

Shaded area = 4*2 - 2* (1/4)π2^2

i.e. Shaded Area = 8 - 2π

Perimeter of shaded Region = Circumference of two quarter circles +AB

i.e. Perimeter of shaded region = 2*(1/4)2π*2 + 4 = 2π+4

Answer: option A

P.S. By mistake I first calculated the area of shaded region which I have deleted on purpose so that reader can also understand how to calculate the area of shaded region in the given case
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Re: In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   16 Mar 2018, 00:08
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Bunuel wrote:

In the figure above, ABCD is a rectangle and DA and CB are radii of the circles shown. If AB = 4, what is the perimeter of the shaded region?

(A) 2π + 4

(B) 4π + 4

(C) 4π + 8

(D) 8π + 8

(E) 8π + 16


Show Spoiler
Attachment:
2018-03-16_1011_004.png



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In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   16 Mar 2018, 01:40
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2018-03-16_1011_004.png
2018-03-16_1011_004.png [ 11.16 KiB | Viewed 6196 times ]


In order to find the perimeter of the shaded region, we need to know the
length of the two curved portions and add it to the length of AB,which is 4.

Let the radii of the two circles be x and y.
From the above diagram, x+y = 4.

The perimeter of the curved portions can be calculated using the formula
for the length of the sector which is \(\frac{@}{360} * 2 * \pi *\) radius (Here @=90)

Sum of the perimeter(of the curved surfaces) is \(\frac{90}{360} * 2 * \pi * x + \frac{90}{360} * 2 * \pi * y\)

This can be further simplified as \(\frac{1}{4}*2*\pi*(x+y) = \frac{1}{2}*\pi*(4) = 2*\pi\)

Therefore, the perimeter of the shaded region is \(2\pi + 4\) (Option A)
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Re: In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   06 Apr 2018, 12:28
I am able to establish that you need to find the area of the rectangle and subtract 1/2 the area of the circle, but since we aren't given the radius of the circle I don't understand how this is possible to determine. What am I missing?
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Re: In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   06 Apr 2018, 12:40
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lostnumber wrote:
I am able to establish that you need to find the area of the rectangle and subtract 1/2 the area of the circle, but since we aren't given the radius of the circle I don't understand how this is possible to determine. What am I missing?


I think solutions above make it clear: we are given that AB = 4.

AB = 2* Radius of each circle, so radius = 2.
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Re: In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   06 Apr 2018, 13:01
Bunuel wrote:
lostnumber wrote:
I am able to establish that you need to find the area of the rectangle and subtract 1/2 the area of the circle, but since we aren't given the radius of the circle I don't understand how this is possible to determine. What am I missing?


I think solutions above make it clear: we are given that AB = 4.

AB = 2* Radius of each circle, so radius = 2.


I looked over more carefully and EgmatQuantExpert's answer gave me what I needed with the diagram he included. I did not understand how we knew AB = 2R, but seeing it drawn out it makes sense now. I should have ready more carefully, sorry!
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Re: In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   06 Apr 2018, 21:02
[quote="Bunuel"]
In the figure above, ABCD is a rectangle and DA and CB are radii of the circles shown. If AB = 4, what is the perimeter of the shaded region?

(A) 2π + 4

(B) 4π + 4

(C) 4π + 8

AB=CD=2*r=4
Hence r =2
Perimeter for 90 curvature is given by =(2*pi*r)*(90/360)
=pi*r/2
Total perimeter for shaded part= AB +2*perimeter of curvature
=4+2pi


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Re: In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   07 Apr 2018, 20:57
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lostnumber wrote:
Bunuel wrote:
lostnumber wrote:
I am able to establish that you need to find the area of the rectangle and subtract 1/2 the area of the circle, but since we aren't given the radius of the circle I don't understand how this is possible to determine. What am I missing?


I think solutions above make it clear: we are given that AB = 4.

AB = 2* Radius of each circle, so radius = 2.


I looked over more carefully and EgmatQuantExpert's answer gave me what I needed with the diagram he included. I did not understand how we knew AB = 2R, but seeing it drawn out it makes sense now. I should have ready more carefully, sorry!


We are glad that you identified your mistake and rectified it by yourself.

I want to add some points just to give you some more understanding.

Since AD and BC are the radii of, let's say circle C1 and circle C2.

Hence, D is the centre of C1 and C is the centre of C2. And,
P also extends from the centre of the C1 i.e. D till P, the circumference of C1. Hence It is also the radius of C1.
In the similar fashion, CP is the radius of the circle C2.

I hope this makes things more clear for you.
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Re: In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   07 Apr 2018, 21:44
1. Find length of the arc
2. 2* length of the arc
3. Add this to length of the AB.
Now length of AB is 4 , hence length of CD=4 . Radius of each circle is CD/2.
Length of the arc = Theta/360 * 2*pi*r , theta here is 90 degrees since ABCD is rectangle
The answer is A
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Re: In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   09 Apr 2018, 07:06
EgmatQuantExpert wrote:

We are glad that you identified your mistake and rectified it by yourself.

I want to add some points just to give you some more understanding.

Since AD and BC are the radii of, let's say circle C1 and circle C2.

Hence, D is the centre of C1 and C is the centre of C2. And,
P also extends from the centre of the C1 i.e. D till P, the circumference of C1. Hence It is also the radius of C1.
In the similar fashion, CP is the radius of the circle C2.

I hope this makes things more clear for you.


Based on the information given in the question, is it mathematically necessary that the circles are the same size, or is that an assumption we are making based on the diagram? I remember back in highschool math our teacher would trick us all the time by giving figures the visually looked like they were symetrical or congruent but mathematically weren't. His point was that you can't assume any information not given in the problem. Is there some mathematical reason why these circles could not be different sizes?
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Re: In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   09 Apr 2018, 07:08
Actually, I've thought about it, and I realized that they must be the same size because we are told that AD and BC are the radius of the respective circles. Thus we know that points D and C must be the center of the circles and since a rectangle is symmetrical everything must be the same.

But as a larger point, will the GMAT ever ask trick questions such as my highschool math teacher did? IE give a diagram where it looks like something is a 90 degree angle but it isn't, or two figures look like the same size but aren't, etc.
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Re: In the figure above, ABCD is a rectangle and DA and CB are radii of [#permalink] New post   28 Oct 2022, 23:07
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