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# If the above figure is a square attached to a semicircle, and the diag

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Math Expert
Joined: 02 Sep 2009
Posts: 55231
If the above figure is a square attached to a semicircle, and the diag  [#permalink]

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13 Sep 2018, 04:58
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Difficulty:

45% (medium)

Question Stats:

37% (01:43) correct 63% (01:33) wrong based on 38 sessions

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If the above figure is a square attached to a semicircle, and the diagonal of the square is 12 what is the area of the whole figure?

A. $$36 + 9\pi$$

B. $$72 + 9\pi$$

C. $$72 + 18\pi$$

D. $$144 + 18\pi$$

E. $$144 + 36\pi$$

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Re: If the above figure is a square attached to a semicircle, and the diag  [#permalink]

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13 Sep 2018, 07:13
Imo C
Diagonal is 12 that means the side of the square is 6sqrt2. The area of the square is 72.
Now the side of the square forms the diameter of the circle therefore the radius of the circle is 3sqrt2.
Area of the semicircle is 18π.
72+18π
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Re: If the above figure is a square attached to a semicircle, and the diag  [#permalink]

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13 Sep 2018, 08:48
IMO, This problem is best solved with estimation and logic.

Answer choice A implies that the side of the square is 6.
B and C imply that the side of the square is between 6 and 12.
D and E imply that the side of the square is 12.

Which makes the most sense? D and E are out because the diagonal and the side can't be the same length.
A is out because if a side is 6, two sides are 12 and would form a straight line the same length as the diagonal.

Side note here, the two shorter sides of a triangle must total a distance longer than the longest side. The diagonal and the two sides form a triangle, therefore....

So it's either B or C.

The side of the square is the diameter of the circle. The area is found by squaring the radius, which is half of the diameter. If the diameter is greater than 6 and less than 12, the radius is greater than 3 but less than 6. B implies a radius that equals 3, so eliminate B.

C is correct.
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If the above figure is a square attached to a semicircle, and the diag  [#permalink]

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13 Sep 2018, 09:05
jaysonbeatty12 wrote:
IMO, This problem is best solved with estimation and logic.

Answer choice A implies that the side of the square is 6.
B and C imply that the side of the square is between 6 and 12.
D and E imply that the side of the square is 12.

Which makes the most sense? D and E are out because the diagonal and the side can't be the same length.
A is out because if a side is 6, two sides are 12 and would form a straight line the same length as the diagonal.

Side note here, the two shorter sides of a triangle must total a distance longer than the longest side. The diagonal and the two sides form a triangle, therefore....

So it's either B or C.

The side of the square is the diameter of the circle. The area is found by squaring the radius, which is half of the diameter. If the diameter is greater than 6 and less than 12, the radius is greater than 3 but less than 6. B implies a radius that equals 3, so eliminate B.

C is correct.

Why the side of the square cannot be 6, if the diagonal is 6 r square 2?

Thank you.

Edit: Don't worry I've seen my mistake.
If the above figure is a square attached to a semicircle, and the diag   [#permalink] 13 Sep 2018, 09:05
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