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Director  Joined: 09 Aug 2006
Posts: 592
A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 39% (01:47) correct 61% (02:02) wrong based on 342 sessions

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A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01
II. 2.00
III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only
Math Expert V
Joined: 02 Sep 2009
Posts: 58396
Re: PS: Area of Rectangle  [#permalink]

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8
5
voodoochild wrote:
walker wrote:
Area of a rectangle ranges from 0 to area of square=2. S0, I and II is possible.

Bunuel / Walker,
I have a question. If the diameter = 2 => diagonal of square =2 => 2(side of a square)^2 = 2 => (side of a square)^2 = 1.

Hence, the area of square is 1!

Thoughts?

If the length of a diagonal of a square is 2, then $$side^2+side^2= diagonal^2$$ --> $$2*side^2=2^2$$ --> $$side^2=area=2$$.

You could get the side in another way: since the angle between a diagonal and a side in a square is 45 degrees, then $$side=\frac{diagonal}{\sqrt{2}}$$ (from the properties of 45-45-90 triangle).

You could also get the area directly: $$area_{square}=\frac{diagonal^2}{2}=2$$.

Complete solution:
A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01
II. 2.00
III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Look at the diagram below:
Attachment: m24-05.png [ 10.63 KiB | Viewed 8463 times ]

If the width of blue rectangle is small enough then its area could be 0.01.

Generally, the are of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle).

Now, since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is $$0<area\leq{2}$$. So, I and II are possible values of the area. (Else you can notice that the area of the circle is $$\pi{r^2}=\pi\approx{3.14}$$ and the area of the inscribed rectangle cannot be greater than that, so III is not possible)

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GMAT 1: 750 Q50 V40 Re: PS: Area of Rectangle  [#permalink]

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2
Area of a rectangle ranges from 0 to area of square=2. S0, I and II is possible.
Attachments t57927.gif [ 3.42 KiB | Viewed 8780 times ]

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Director  Joined: 09 Aug 2006
Posts: 592
Re: PS: Area of Rectangle  [#permalink]

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walker wrote:
Area of a rectangle ranges from 0 to area of square=2. S0, I and II is possible.

Simple enough now! Thanks a bunch Walker.
Director  Joined: 22 Nov 2007
Posts: 893
Re: PS: Area of Rectangle  [#permalink]

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GK_Gmat wrote:
walker wrote:
Area of a rectangle ranges from 0 to area of square=2. S0, I and II is possible.

Simple enough now! Thanks a bunch Walker.

moreover, the area of the circle is 3,14...thus 3,20 is the area of a figure which is not inscribed....
Manager  Joined: 16 Feb 2011
Posts: 164
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Re: PS: Area of Rectangle  [#permalink]

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walker wrote:
Area of a rectangle ranges from 0 to area of square=2. S0, I and II is possible.

Bunuel / Walker,
I have a question. If the diameter = 2 => diagonal of square =2 => 2(side of a square)^2 = 2 => (side of a square)^2 = 1.

Hence, the area of square is 1!

Thoughts?
Senior Manager  Joined: 15 Jun 2010
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WE 1: 7 Yrs in Automobile (Commercial Vehicle industry)
Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

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diagonal of Square = a* root 2=2, Hence a (side of sq)= root 2.

Area of Square = root 2 X root 2 = 2
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Director  Joined: 22 Mar 2011
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Re: PS: Area of Rectangle  [#permalink]

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voodoochild wrote:
walker wrote:
Area of a rectangle ranges from 0 to area of square=2. S0, I and II is possible.

Bunuel / Walker,
I have a question. If the diameter = 2 => diagonal of square =2 => 2(side of a square)^2 = 2 => (side of a square)^2 = 1.

Hence, the area of square is 1!

Thoughts?

Area of a square is also diagonal * diagonal / 2, so in our case is 2 * 2 / 2 =2.

If you want to find the side of the square, then don't forget to square: 2(side of a square)^2 = 2^2=4...

Be more careful with your computations. On the real test you won't get help from the forum's members )
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Director  Joined: 22 Mar 2011
Posts: 588
WE: Science (Education)
Re: PS: Area of Rectangle  [#permalink]

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Bunuel wrote:
voodoochild wrote:
walker wrote:
Area of a rectangle ranges from 0 to area of square=2. S0, I and II is possible.

Bunuel / Walker,
I have a question. If the diameter = 2 => diagonal of square =2 => 2(side of a square)^2 = 2 => (side of a square)^2 = 1.

Hence, the area of square is 1!

Thoughts?

If the length of a diagonal of a square is 2, then $$side^2+side^2= diagonal^2$$ --> $$2*side^2=2^2$$ --> $$side^2=area=2$$.

You could get the side in another way: since the angle between a diagonal and a side in a square is 45 degrees, then $$side=\frac{diagonal}{\sqrt{2}}$$ (from the properties of 45-45-90 triangle).

You could also get the area directly: $$area_{square}=\frac{diagonal^2}{2}=2$$.

Complete solution:
A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01
II. 2.00
III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Look at the diagram below: If the width of blue rectangle is small enough then its area could be 0.01.

Generally, the are of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle).

Now, since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is $$0<area\leq{2}$$. So, I and II are possible values of the area. (Else you can notice that the area of the circle is $$\pi{r^2}=\pi\approx{3.14}$$ and the area of the inscribed rectangle cannot be greater than that, so III is not possible)

The link to the figure is not working/missing...

Did you mean something like this? (See the attached figure)

We can make the area of the inscribed rectangle as close as possible to 0, when for example the length is getting closer, and closer to the diameter, while the width is getting closer and closer to 0.

The maximum area can be obtained when the height of the right triangle, which is half of the rectangle and it is inscribed in the half circle, is equal to the radius of the circle. In this case, the rectangle becomes a square.
Attachments MAxRectangleAreaInCircle.jpg [ 24.29 KiB | Viewed 8445 times ]

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Math Expert V
Joined: 02 Sep 2009
Posts: 58396
Re: PS: Area of Rectangle  [#permalink]

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EvaJager wrote:
The link to the figure is not working/missing...

It should be visible now.
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Manager  Joined: 16 Feb 2011
Posts: 164
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Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

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Thanks Bunuel...I missed this problem because of a silly mistake. <sad> Thanks again.
Senior Manager  Joined: 13 May 2013
Posts: 398
Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

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2
A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

If the circle has a diameter of 2, then it's radius is 1 so it's area is pi*1^2 or simply pi. Pi is equal to 3.1415. A a rectangle of .01 or 2.0 could fit inside this circle but not a rectangle with an area larger than the circle.

D

I. 0.01
II. 2.00
III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only
Intern  Joined: 03 Mar 2013
Posts: 4
Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

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Rectangle could be a square in this problem? The correct answer is still D, but I'm just curious
Math Expert V
Joined: 02 Sep 2009
Posts: 58396
Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

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AlexeyGevlich wrote:
Rectangle could be a square in this problem? The correct answer is still D, but I'm just curious

Yes, because all squares are rectangles.
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Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

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1
GK_Gmat wrote:
A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01
II. 2.00
III. 3.20

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

Let $$x$$ be the length & $$y$$ be the width of the rectangle. Its diagonal will be the diameter of the circle, as the rectangle is inscribed in the circle.

Therefore length of diagonal = $$2$$ & Area of the rectangle = $$xy$$

Hence we have $$x^2 + y^2 = 4$$

or $$(x - y)^2 + 2xy = 4$$

or $$(x - y)^2 = 4 - 2xy$$

LHS has to be positive or 0, hence $$4 - 2xy >=0$$

We can see that only I & II satisfy the above inequality.

Thanks,
GyM
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Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

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