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A is the center of the circle, and the length of AB is . The blue shad

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A is the center of the circle, and the length of AB is . The blue shad  [#permalink]

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New post 20 Mar 2015, 07:07
1
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A
B
C
D
E

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A is the center of the circle, and the length of AB is \(4\sqrt{2}\). The blue shaded region is a square. What is the area of the shaded region?

A. 4(4 - π)
B. 4(8 - π)
C. 8(2 - π)
D. 8(8 - π)
E. 16(4 - π)

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Re: A is the center of the circle, and the length of AB is . The blue shad  [#permalink]

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New post 20 Mar 2015, 12:37
1
correct choice is E

side of square =2r

4sqt2=r *sqt2

r=4

so we can calculate the area of square and circle and minus
which boils down to option E for the correct answer
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Re: A is the center of the circle, and the length of AB is . The blue shad  [#permalink]

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New post 20 Mar 2015, 12:54
1
consider radius of circle = r
so side of square = 2r

r^2+r^2 = {4sqrt(2)}^2
2r^2 = 32
r^2 = 16

so area of circle = pi*r^2 => pi*16
area of square = 4*r^2 = 4*16
so area of shaded region == area of square - area of cirlce == 16(4-pi)
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A is the center of the circle, and the length of AB is . The blue shad  [#permalink]

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New post Updated on: 21 Mar 2015, 03:07
2
Since AB is half the diagonal of the square, the diagonal is \(8\sqrt{2}\), which is \(\sqrt{2}\) the side of the square. The side of the square is then \(\frac{8\sqrt{2}}{\sqrt{2}}=8\).

The area of the square is then \(8^2=64\).

The side of the square equals twice the radium of the circle. Therefore, the area of the circle is \(\pi\cdot4^2=16\pi\)

The shaded area is \(64-16\pi=16(4-\pi)\)

The correct answer is E

Originally posted by ngie on 20 Mar 2015, 13:22.
Last edited by ngie on 21 Mar 2015, 03:07, edited 1 time in total.
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Re: A is the center of the circle, and the length of AB is . The blue shad  [#permalink]

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New post 20 Mar 2015, 20:45
Bunuel wrote:
Attachment:
areashade_q2.png
A is the center of the circle, and the length of AB is \(4\sqrt{2}\). The blue shaded region is a square. What is the area of the shaded region?

A. 4(4 - π)
B. 4(8 - π)
C. 8(2 - π)
D. 8(8 - π)
E. 16(4 - π)

Kudos for a correct solution.


The line segment AB forms a 45 45 90 triangle with a hypotenuse of \(4\sqrt{2}\)
As a result both a side of the square and the radius is 4.
8^2 - (4^2)π
16(4 - π)

Answer: E
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Re: A is the center of the circle, and the length of AB is . The blue shad  [#permalink]

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New post 23 Mar 2015, 05:08
Bunuel wrote:
Image
A is the center of the circle, and the length of AB is \(4\sqrt{2}\). The blue shaded region is a square. What is the area of the shaded region?

A. 4(4 - π)
B. 4(8 - π)
C. 8(2 - π)
D. 8(8 - π)
E. 16(4 - π)

Kudos for a correct solution.


MAGOOSH OFFICIAL SOLUTION:
Attachment:
areashade_explanation.png
areashade_explanation.png [ 46.82 KiB | Viewed 4959 times ]

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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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A is the center of the circle, and the length of AB is . The blue shad  [#permalink]

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New post 26 Mar 2015, 19:32
There is a direct relation between area of inscribed circle & area of square

\(Area of circle = Area of square * \frac{\pi}{4}\)

Diagonal of square \(= 2 * 4\sqrt{2} = 8\sqrt{2}\)

Side of square \(= \sqrt{\frac{(8\sqrt{2})^2}{2}} = 8\)

Area of square \(= 8^2 = 64\)

Area of circle \(= 64 * \frac{\pi}{4} = 16\pi\)

Area of shaded region \(= 64 - 16\pi = 16(4-\pi)\)
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Re: A is the center of the circle, and the length of AB is . The blue shad  [#permalink]

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New post 26 Mar 2015, 22:59
Hi All,
Area of the shaded region = Area of the square – area of the circle.
Since line segment AB forms the 45-45-90 triangle with square.
It gives side of the square as 8. And radius of the circle as 4.
Area of the shaded region = 64 – 16pi.
= 16(4-Pi).
You can also approach this question using POE.
the shaded region should be slightly less than 1/4th of the area of the square. Since area of the square is 64. We are looking some answer close to 16.
Let pi = 3.1
A. 4(4 - π) this is less than 4 eliminate.
B. 4(8 - π) this is more than 16 eliminate.
C. 8(2 - π) this is negative eliminate.
D. 8(8 - π) this is more than 16 eliminate.
E. 16(4 - π) this is very close to 16. So answer is E.
POE is just in case if you don’t know how to solve.
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Re: A is the center of the circle, and the length of AB is . The blue shad  [#permalink]

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New post 02 Mar 2019, 05:32
Bunuel wrote:
Attachment:
areashade_q2.png
A is the center of the circle, and the length of AB is \(4\sqrt{2}\). The blue shaded region is a square. What is the area of the shaded region?

A. 4(4 - π)
B. 4(8 - π)
C. 8(2 - π)
D. 8(8 - π)
E. 16(4 - π)

Kudos for a correct solution.


okay

since the length of AB is 4√2, the diagonal of the square is 8√2, and thus each side of the square measures 8 and radius of the circle measures 4

now the shaded area = 64 - pi 4^2 = 16(4-pi) = E the answer

thanks
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Re: A is the center of the circle, and the length of AB is . The blue shad   [#permalink] 02 Mar 2019, 05:32
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