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In a rectangular coordinate system, point A has coordinates  [#permalink]

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In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

A. $$\frac{\pi}{\pi+\sqrt{2}}$$

B. $$\frac{\pi}{\pi+1}$$

C. $$\sqrt{\frac{2}{\pi}}$$

D. $$\frac{\pi+2}{2\pi}$$

E. $$\frac{2\pi}{2\pi+1}$$

Originally posted by Financier on 20 Aug 2010, 14:30.
Last edited by Bunuel on 19 Apr 2012, 11:53, edited 1 time in total.
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Re: What fraction of the area of circle C lies within ...  [#permalink]

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Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Look a the diagram:
Attachment: CS.jpg [ 22.42 KiB | Viewed 13792 times ]
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is $$C=\pi{r^2}=\pi$$;
Area of a square is half of the product of diagonals, as diagonal equals to $$2r=2$$, then $$S=\frac{2^2}{2}=2$$;

Area of a circle without the red parts is $$C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}$$;

Ratio of the are of this region to area of a circle is $$\frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}$$.

Hope it's clear.
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Re: What fraction of the area of circle C lies within ...  [#permalink]

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Ok so I am typing this on my mobile with imagination... Imagine a square of side 1 and a circle circumscribed.. Qn is what is the area that is lying on the 2 bulging sides..
Area of square = 1
area of circle = pi/2

The area of 4 bulging sides = (pi/2-1)
Fraction of circle area inside 1st Q= [pi/2-1/2(pi/2-1)]/pi/2 = (pi/2-pi/4+1/2)/(pi/2) = (pi/4+1/2)/(pi/2) = (pi+2)/2pi

Posted from my mobile device
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Originally posted by mainhoon on 20 Aug 2010, 14:48.
Last edited by mainhoon on 20 Aug 2010, 16:05, edited 2 times in total.
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Re: What fraction of the area of circle C lies within ...  [#permalink]

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Let us assume A to be (1,1).Then, distance from origin = sqrt 2 = Diagonal of square = Diameter of circle.

Area of square = 1 sq units and
Area of circle = Pi r^2 = Pi (sqrt 2 / 2)^2 = Pi / 2

Area of 4 bulging sides = (Pi/2) - 1 = (Pi - 2) / 2 and Area of 2 bulging side = (Pi - 2) / 4.

Therefore, required fraction = Area of 2 bulging side / Area of circle = [(Pi - 2) / 4] / (Pi/2) = Pi -2 / 2Pi.

But answer posted by Bunuel is : Pi + 2 / 2Pi and I take his explainations as absolute. Since I got a different answer....please explain where I am wrong.
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Re: What fraction of the area of circle C lies within ...  [#permalink]

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Area of circle = pie(r^2)
diagonal of square = 2^1/2*side
2^1/2*side = 2r
side = r*2^1/2
area of square = 2(r^2)

Area of red portion = Area of Circle - Area of square /2 = pie-2/2pie

Pls correct me if wrong
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Re: What fraction of the area of circle C lies within ...  [#permalink]

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GMATD11 wrote:
Area of circle = pie(r^2)
diagonal of square = 2^1/2*side
2^1/2*side = 2r
side = r*2^1/2
area of square = 2(r^2)

Area of red portion = Area of Circle - Area of square /2 = pie-2/2pie

Pls correct me if wrong

Solution: in-a-rectangular-coordinate-system-point-a-has-coordinates-99510.html#p767285
OA: D.
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Re: In a rectangular coordinate system, point A has coordinates  [#permalink]

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Hi, Cant we do this by calculating the area of the sector and then the area of the circle??? Pls help me understand!
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Re: In a rectangular coordinate system, point A has coordinates  [#permalink]

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es we can do it by taking any value for (d,d) and calculating the area deducting the sector.
Take (d,d) as 2,2. the dia meter becomes 2 root 2 and radius root2
so area of circle = 2pie
area of 2 sectors= 2*(90/360)*2 pie = pie
But have to add another half of square = 1/2 * 4 = 2
So area in 1st quadrant = 2pie-pie+2= pie+2
and area of circle = 2pie
So ratio= (2+pie)/2pie
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Re: What fraction of the area of circle C lies within ...  [#permalink]

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Bunuel wrote:
Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Look a the diagram:
Attachment:
CS.jpg
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is $$C=\pi{r^2}=\pi$$;
Area of a square is half of the product of diagonals, as diagonal equals to $$2r=2$$, then $$S=\frac{2^2}{2}=2$$;

Area of a circle without the red parts is $$C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}$$;

Ratio of the are of this region to area of a circle is $$\frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}$$.

Hope it's clear.

Hi,

Area of a circle without the red parts is $$C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}$$;

is already describing the circle's portion, which is available in 1st Q. why we have again check for the ratio b/w this and circle again.I got this doubt since question has asked us to find the fraction of the area of circle C lies within the first quadrant.

Kindly help me understand ..
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Re: What fraction of the area of circle C lies within ...  [#permalink]

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FTG wrote:
Bunuel wrote:
Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Look a the diagram:
Attachment:
CS.jpg
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is $$C=\pi{r^2}=\pi$$;
Area of a square is half of the product of diagonals, as diagonal equals to $$2r=2$$, then $$S=\frac{2^2}{2}=2$$;

Area of a circle without the red parts is $$C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}$$;

Ratio of the are of this region to area of a circle is $$\frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}$$.

Hope it's clear.

Hi,

Area of a circle without the red parts is $$C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}$$;

is already describing the circle's portion, which is available in 1st Q. why we have again check for the ratio b/w this and circle again.I got this doubt since question has asked us to find the fraction of the area of circle C lies within the first quadrant.

Kindly help me understand ..

The question asks for "fraction" or "proportion" of the circle. (e.g. say 1/2 or 3/5th of circle lies in the first quadrant)
Hence you need to take the ratio of Area (in 1st quadrant) to Total area to find the fraction.
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Re: What fraction of the area of circle C lies within ...  [#permalink]

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Bunuel wrote:
Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Look a the diagram:
Attachment:
CS.jpg
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is $$C=\pi{r^2}=\pi$$;
Area of a square is half of the product of diagonals, as diagonal equals to $$2r=2$$, then $$S=\frac{2^2}{2}=2$$;

Area of a circle without the red parts is $$C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}$$;

Ratio of the are of this region to area of a circle is $$\frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}$$.

Hope it's clear.

Quote:
Area of a square is half of the product of diagonals, as diagonal equals to $$2r=2$$, then $$S=\frac{2^2}{2}=2$$;

is this because you are taking into account the portion of the square which doesnt touch the red parts of the circle? Since area of Square is A = a^2 ?
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Posts: 55732
Re: What fraction of the area of circle C lies within ...  [#permalink]

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pharm wrote:
Bunuel wrote:
Financier wrote:
In a rectangular coordinate system, point A has coordinates (d, d), where d > 0. Point A and the origin form the endpoints of a diameter of circle C. What fraction of the area of circle C lies within the first quadrant?

Look a the diagram:
Attachment:
CS.jpg
We have circle with radius=1 and square inscribed in it. Now imagine the base of square to be x-axis and left side of square to be y-axis. We need to find the ratio of area of a circle without the red parts to the area of whole circle.

Area of a circle is $$C=\pi{r^2}=\pi$$;
Area of a square is half of the product of diagonals, as diagonal equals to $$2r=2$$, then $$S=\frac{2^2}{2}=2$$;

Area of a circle without the red parts is $$C-\frac{C-S}{2}=\pi-\frac{\pi-2}{2}=\frac{\pi+2}{2}$$;

Ratio of the are of this region to area of a circle is $$\frac{\frac{\pi+2}{2}}{\pi}=\frac{\pi+2}{2\pi}$$.

Hope it's clear.

Quote:
Area of a square is half of the product of diagonals, as diagonal equals to $$2r=2$$, then $$S=\frac{2^2}{2}=2$$;

is this because you are taking into account the portion of the square which doesnt touch the red parts of the circle? Since area of Square is A = a^2 ?

If I understand correctly you are asking about the area of a square: $$area_{square}=side^2=\frac{diagonal^2}{2}$$. This is a general formula for the area of any square.

Hope it helps.
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Re: In a rectangular coordinate system, point A has coordinates  [#permalink]

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Yea, thank you cleared things up
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Re: In a rectangular coordinate system, point A has coordinates  [#permalink]

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pharm wrote:
Yea, thank you cleared things up

Forgot to mention that the area of a rhombus is also equals to half of the product of diagonals: $$area_{rhombus}=\frac{d_1*d_2}{2}$$, where $$d_1$$ and $$d_2$$ are the lengths of the diagonals (or $$bh$$, where $$b$$ is the length of the base and $$h$$ is the altitude).
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In a rectangular coordinate system, point A has coordinates  [#permalink]

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why we are considering square for this question, its nowhere specified in the question. I do not get this at all, please help.

Regards
Megha
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Re: In a rectangular coordinate system, point A has coordinates  [#permalink]

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A quick look at the answer choices tells you that the ratio does not depend on the value of "d". Hence you can choose "smart numbers" for d - For example 2. Which makes the radius of the circle sqrt(2). The all we need to do is add the area of the semi circle with the area of the triangle to get the area in the first quadrant (refer the diagram by Bunuel).

Solve and the answer is D.
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Re: In a rectangular coordinate system, point A has coordinates  [#permalink]

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megha_2709 wrote:
why we are considering square for this question, its nowhere specified in the question. I do not get this at all, please help.

Regards
Megha

If you drop perpendiculars to x & y axes from (d,d) you will get (d,0) and (0,d). These two points are "d" units away from the origin as well as from (d,d). Knowing that all sides are of the same length and that the 3 angles are 90 degrees, (we drew perpendiculars to the axes and the axes themselves are at a () degree angle) we can conclude that the figure is a square.

Hope it helps.
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Re: In a rectangular coordinate system, point A has coordinates  [#permalink]

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