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# In the quarter circle above, what is the area of the shaded region?

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Math Expert
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In the quarter circle above, what is the area of the shaded region?  [#permalink]

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17 Sep 2018, 21:20
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Difficulty:

95% (hard)

Question Stats:

43% (01:20) correct 57% (01:34) wrong based on 97 sessions

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In the quarter circle above, what is the area of the shaded region?

(1) The perimeter of the rectangle ABDF is 14.
(2) AC is 5 and AF is 3

Attachment:

image011 (1).jpg [ 2.48 KiB | Viewed 843 times ]

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Re: In the quarter circle above, what is the area of the shaded region?  [#permalink]

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18 Sep 2018, 17:38
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Afc0892 wrote:
Bunuel wrote:

In the quarter circle above, what is the area of the shaded region?

(1) The perimeter of the rectangle ABDF is 14.
(2) AC is 5 and AF is 3

Attachment:
The attachment image011 (1).jpg is no longer available

Bunuel, How's the answer B and not C?

Sir, Initially i marked C but answer should be B. I have attached the calculation. Please check.
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IMG_20180919_070357.jpg [ 2.07 MiB | Viewed 637 times ]

##### General Discussion
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Re: In the quarter circle above, what is the area of the shaded region?  [#permalink]

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17 Sep 2018, 21:28
Image
In the quarter circle above, what is the area of the shaded region?

(1) The perimeter of the rectangle ABDF is 14.
(2) AC is 5 and AF is 3

from stmnt 1 : 2(l+b)= 14
from stmnt 2: find the quarter circle circumference : (2*3.14*AC)*1/4= 2.5 *3.14--- (1)
and also value of l by using stmnt 1 eqn: 2l+ 6= 14; l= 4---->a
using above relation of a we can find diagonal BF which shall be 5 as per triangle (3:4:5)

find area of triangle ABF: viz 6 and subtract it from (1) value

we can find shaded region area...

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Re: In the quarter circle above, what is the area of the shaded region?  [#permalink]

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17 Sep 2018, 21:30
From statement 1:

Perimeter of ABDF = 14 = 2(AB+AF)
Then AB+AF = 7.

No extra info. Hence insufficient.

From statement 2:

AC = 5 = radius of the quarter circle.
AF = 3.
FE becomes 2.
No info about AB. Hence insufficient.

Combining both:

AB becomes 4.

Area of the quarter circle = $$\frac{90}{360}$$*Pi*5^2 = $$\frac{25}{4}$$Pi

Area of the shaded region = Area of the quarter circle-$$\frac{1}{2}$$Area of the rectangle
Area of the shaded region = $$\frac{25}{4}$$Pi-6

Hence sufficient.

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Re: In the quarter circle above, what is the area of the shaded region?  [#permalink]

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17 Sep 2018, 21:57
From statement 1: Perimeter of rectangle is known. With given data we will not be able to find the area of unshaded portion.
From statement 2: radious of quarter circle is known and base of unshaded portion is known. But,we will not be able to find the area of unshaded portion.

Combining both the statements, area of unshaded portion and quarter circle can be found out.

Hence, option C is correct. IMO.

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In the quarter circle above, what is the area of the shaded region?  [#permalink]

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17 Sep 2018, 22:07
In the quarter circle above, what is the area of the shaded region?

(1) The perimeter of the rectangle ABDF is 14.
(2) AC is 5 and AF is 3

(1)--> 2(l+w)=14 --> (l+w)=7 -->clearly insuff. (l and w values not given)
(2)--> AC=5=radius and AF=l=3 --> insuff. (we don't know what value can b can take)

Combining (1) and (2),
(1)--> l+w=7 and (2)-->l=3, we can find b=4;
Since rectangle forms a right angle,
Area of unshaded=1/2 * l * b -->Find the area of BAF
Area of circle = 25pi -->(r=5)

Ans C
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Re: In the quarter circle above, what is the area of the shaded region?  [#permalink]

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18 Sep 2018, 17:22
Bunuel wrote:

In the quarter circle above, what is the area of the shaded region?

(1) The perimeter of the rectangle ABDF is 14.
(2) AC is 5 and AF is 3

Attachment:
image011 (1).jpg

Bunuel, How's the answer B and not C?
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If you are not badly hurt, you don't learn. If you don't learn, you don't grow. If you don't grow, you don't live. If you don't live, you don't know your worth. If you don't know your worth, then what's the point?

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Re: In the quarter circle above, what is the area of the shaded region?  [#permalink]

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18 Sep 2018, 18:43
AnupamKT wrote:
Afc0892 wrote:
Bunuel wrote:
In the quarter circle above, what is the area of the shaded region?

(1) The perimeter of the rectangle ABDF is 14.
(2) AC is 5 and AF is 3

Sir, Initially i marked C but answer should be B. I have attached the calculation. Please check.

Hello, How can AD could be 5?
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Re: In the quarter circle above, what is the area of the shaded region?  [#permalink]

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18 Sep 2018, 18:47
jorgetomas9 wrote:
Hello, How can AD could be 5?

A is the center of the quarter circle. And AD, AE, and AC are all radii. Hence AD is 5.
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In the quarter circle above, what is the area of the shaded region?  [#permalink]

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19 Sep 2018, 03:55
Area of shaded region = area of quarter of the circle - area of triangle

Translated into formulas, we get: $$(1/4)pi*r^2 - (1/2) base*height$$

Thus, the unknowns we need to solve the question are r, base, and height.

(1) This statement does not give us all the unknowns we need. Insufficient
(2) This statement provides us with the radius (AC =5) and the base (AF=3). Now, we also know that the diagonal cuts the rectangle in half and therefore two of its angles are halfed into 45°, making the resulting two triangles into two identical isosceles triangles. Knowing that the ratio of the sides of a right isosceles triangle is x:x:x$$\sqrt{2}$$, we also know that base (AC) = height (AB) =5. Based on this information, we can calculate the area of the quarter circle and the area of the triangle.

Statement (2) is sufficient. Answer B

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In the quarter circle above, what is the area of the shaded region?  [#permalink]

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20 Nov 2018, 19:27
I understand the logic of B, but how can you guarantee ABDF is a rectangle? if BD is not parallel to AF (so assume AB=3 and DF=4), then the diagonals are not equal, leading to C as the correct answer. Can someone please explain how 2) ensures that ABDF is a rectangle on its own?
In the quarter circle above, what is the area of the shaded region? &nbs [#permalink] 20 Nov 2018, 19:27
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