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Rectangle ABCD is inscribed in circle P. What is the area of circle P?

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Rectangle ABCD is inscribed in circle P. What is the area of circle P? [#permalink]

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Rectangle ABCD is inscribed in circle P. What is the area of circle P?

(1) The area of rectangle ABCD is 100.

(2) Rectangle ABCD is a square.


Attachment:
inscribedsquare.jpg
inscribedsquare.jpg [ 13.44 KiB | Viewed 7665 times ]

Originally posted by AccipiterQ on 21 Oct 2013, 08:36.
Last edited by Bunuel on 24 Feb 2018, 00:34, edited 2 times in total.
Edited the question.
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Re: Rectangle ABCD is inscribed in circle P. What is the area of circle P? [#permalink]

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New post 21 Oct 2013, 08:42
AccipiterQ wrote:
Rectangle ABCD is inscribed in circle P. What is the area of circle P?

(1) The area of rectangle ABCD is 100.

(2) Rectangle ABCD is a square.



1. area of rectangle ABCD is 100. If the rectangle is also a square, then you can solve. It doesn't say that though, so the rectangle could have a number of different side lengths. insufficient.

2. rectangle ABCD is a square. No values are given. Insufficient.


1&2 combined. the rectangle is also a square, meaning that all sides are length of 10. you can draw a line from a to c (which would be the cirlce's diameter), and form a right triangle, with two sides of 10, and AC (the hypothaneuse (sp??) being unknown).

10^2+10^2=X^2
200=X^2
X=\(\sqrt{200}\)
X=10*\(\sqrt{2}\)
so the diameter is 10*\(\sqrt{2}\)

since the radius is half the diamter you get 5*\(\sqrt{2}\)

area of a cirlce is pi*r^2, so pi*(5*\(\sqrt{2}\))^2
=pi*25*2
=50*pi

so 1&2 are sufficient.
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Re: Rectangle ABCD is inscribed in circle P. What is the area of circle P? [#permalink]

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New post 21 Oct 2013, 09:14
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1
AccipiterQ wrote:
AccipiterQ wrote:
Image
Rectangle ABCD is inscribed in circle P. What is the area of circle P?

(1) The area of rectangle ABCD is 100.

(2) Rectangle ABCD is a square.



1. area of rectangle ABCD is 100. If the rectangle is also a square, then you can solve. It doesn't say that though, so the rectangle could have a number of different side lengths. insufficient.

2. rectangle ABCD is a square. No values are given. Insufficient.


1&2 combined. the rectangle is also a square, meaning that all sides are length of 10. you can draw a line from a to c (which would be the cirlce's diameter), and form a right triangle, with two sides of 10, and AC (the hypothaneuse (sp??) being unknown).

10^2+10^2=X^2
200=X^2
X=\(\sqrt{200}\)
X=10*\(\sqrt{2}\)
so the diameter is 10*\(\sqrt{2}\)

since the radius is half the diamter you get 5*\(\sqrt{2}\)

area of a cirlce is pi*r^2, so pi*(5*\(\sqrt{2}\))^2
=pi*25*2
=50*pi

so 1&2 are sufficient.


When combined: the area of a square is (diagonal)^2/2, so we are given that (diagonal)^2/2=100 --> \(d=10\sqrt{2}\).
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Re: Rectangle ABCD is inscribed in circle P. What is the area of circle P? [#permalink]

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New post 16 Sep 2015, 00:36
Rectangle ABCD is inscribed in circle P. What is the area of circle P?

(1) The area of rectangle ABCD is 100.

(2) Rectangle ABCD is a square.

Ans is C right for this question ....

as 1 is not sufficient
and 2 alone is also not

But 1 and 2 we get ... square with area 100 so lenght is 10 and we get the diagonal ...half of which is radius of the circle .
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Re: Rectangle ABCD is inscribed in circle P. What is the area of circle P? [#permalink]

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New post 05 Sep 2017, 16:25
AccipiterQ wrote:
Attachment:
inscribedsquare.jpg
Rectangle ABCD is inscribed in circle P. What is the area of circle P?

(1) The area of rectangle ABCD is 100.

(2) Rectangle ABCD is a square.


Ok so Bunuel gave us another trick we can use to attack square within circle problems and circle within square problems-

If you multiply both the diagonals of a square and then divide the product by 2 then you have the area of the square. For example, the area of a 4 x 4 square is 16. The diagonal of the square would then be 4\sqrt{2} - if we square this and divide it by 2 this also results in 16.

Statement 1

What kind of rectangle is this? We cannot use the formula above unless we know that the shape is precisely a square.

Statement 2

We can establish the formula above but with no side lengths we cannot calculate anything.

Statement 1 and 2

Simply do
(x)^2/2= 100
x^2= 50

C
Re: Rectangle ABCD is inscribed in circle P. What is the area of circle P?   [#permalink] 05 Sep 2017, 16:25
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