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Given a rectangle and circle that share a center with a vertex of a tr

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Math Expert
Joined: 02 Sep 2009
Posts: 55266
Given a rectangle and circle that share a center with a vertex of a tr  [#permalink]

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12 Sep 2018, 00:13
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Difficulty:

45% (medium)

Question Stats:

70% (01:57) correct 30% (01:43) wrong based on 24 sessions

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Given a rectangle and circle that share a center with a vertex of a triangle as shown above, what is the area of the shaded region?

(1) Each side of the inscribed rectangle is 2.
(2) The ratio of the area of the rectangle to the area of the circle is d^2 to 2πr^2, where d is the diameter of the circle and r is the radius.

Attachment:

image016.jpg [ 2.22 KiB | Viewed 444 times ]

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Joined: 20 Feb 2015
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Concentration: Strategy, General Management
Given a rectangle and circle that share a center with a vertex of a tr  [#permalink]

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12 Sep 2018, 09:21
Bunuel wrote:

Given a rectangle and circle that share a center with a vertex of a triangle as shown above, what is the area of the shaded region?

(1) Each side of the inscribed rectangle is 2.
(2) The ratio of the area of the rectangle to the area of the circle is d^2 to 2πr^2, where d is the diameter of the circle and r is the radius.

Attachment:
image016.jpg

(1) Each side of the inscribed rectangle is 2.
that means the rectangle is a square
now,
the radius of the circle =$$\frac{diagonal of the square}{2}$$= $$\sqrt{2}$$

we now have all the three sides of the triangle ,$$\sqrt{2}$$,$$\sqrt{2}$$,2

area of the sector =$$\pi r^2$$ @/360
r=$$\sqrt{2}$$
the diagonals of a square intersect at 90 degrees .
therefore @ = 90

sufficient

(2) The ratio of the area of the rectangle to the area of the circle is d^2 to 2πr^2, where d is the diameter of the circle and r is the radius.
we get the value as 7/11
insufficient

A
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Joined: 02 Aug 2015
Posts: 155
Given a rectangle and circle that share a center with a vertex of a tr  [#permalink]

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12 Sep 2018, 11:05
Bunuel wrote:

Given a rectangle and circle that share a center with a vertex of a triangle as shown above, what is the area of the shaded region?

(1) Each side of the inscribed rectangle is 2.
(2) The ratio of the area of the rectangle to the area of the circle is d^2 to 2πr^2, where d is the diameter of the circle and r is the radius.

Attachment:
image016.jpg

Statement 1 - If each side of a rectangle is equal, then it's a square. Given the sides of square=2, so radius of circle will be 2root(2).

Area of shaded region = (Area of circle - Area of square) /4. Hence sufficient.

Statement 2 - Simplifying the equation revealed that the rectangle is a square but no dimensions are given. Hence not sufficient.

Hence A.

Cheers!
Given a rectangle and circle that share a center with a vertex of a tr   [#permalink] 12 Sep 2018, 11:05
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