GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 17 Oct 2019, 16:03

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# A rectangle is inscribed in a circle of diameter 2. Which of

Author Message
TAGS:

### Hide Tags

Director
Joined: 09 Aug 2006
Posts: 599
A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

### Show Tags

04 Jan 2008, 00:23
3
17
00:00

Difficulty:

95% (hard)

Question Stats:

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

### HideShow timer Statistics

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
Joined: 02 Sep 2009
Posts: 58402
Re: PS: Area of Rectangle  [#permalink]

### Show Tags

03 Aug 2012, 07:17
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 8434 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)

_________________
##### General Discussion
CEO
Joined: 17 Nov 2007
Posts: 3071
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Re: PS: Area of Rectangle  [#permalink]

### Show Tags

04 Jan 2008, 01:54
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 8751 times ]

_________________
HOT! GMAT Club Forum 2020 | GMAT ToolKit 2 (iOS) - The OFFICIAL GMAT CLUB PREP APPs, must-have apps especially if you aim at 700+ | Limited Online GMAT/GRE Math tutoring
Senior Manager
Joined: 13 May 2013
Posts: 399
Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

### Show Tags

12 Dec 2013, 12:45
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
Director
Joined: 09 Aug 2006
Posts: 599
Re: PS: Area of Rectangle  [#permalink]

### Show Tags

04 Jan 2008, 03:27
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: 902
Re: PS: Area of Rectangle  [#permalink]

### Show Tags

04 Jan 2008, 06:38
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
Schools: ABCD
Re: PS: Area of Rectangle  [#permalink]

### Show Tags

03 Aug 2012, 07:02
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
Posts: 276
Schools: IE'14, ISB'14, Kellogg'15
WE 1: 7 Yrs in Automobile (Commercial Vehicle industry)
Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

### Show Tags

03 Aug 2012, 07:13
diagonal of Square = a* root 2=2, Hence a (side of sq)= root 2.

Area of Square = root 2 X root 2 = 2
_________________
Regards
SD
-----------------------------
Press Kudos if you like my post.
Debrief 610-540-580-710(Long Journey): http://gmatclub.com/forum/from-600-540-580-710-finally-achieved-in-4th-attempt-142456.html
Director
Joined: 22 Mar 2011
Posts: 588
WE: Science (Education)
Re: PS: Area of Rectangle  [#permalink]

### Show Tags

03 Aug 2012, 07:15
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 )
_________________
PhD in Applied Mathematics
Love GMAT Quant questions and running.
Director
Joined: 22 Mar 2011
Posts: 588
WE: Science (Education)
Re: PS: Area of Rectangle  [#permalink]

### Show Tags

03 Aug 2012, 08:06
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 8416 times ]

_________________
PhD in Applied Mathematics
Love GMAT Quant questions and running.
Math Expert
Joined: 02 Sep 2009
Posts: 58402
Re: PS: Area of Rectangle  [#permalink]

### Show Tags

03 Aug 2012, 08:21
EvaJager wrote:
The link to the figure is not working/missing...

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

### Show Tags

03 Aug 2012, 08:45
Thanks Bunuel...I missed this problem because of a silly mistake. <sad> Thanks again.
Intern
Joined: 03 Mar 2013
Posts: 4
Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

### Show Tags

13 Dec 2013, 10:34
Rectangle could be a square in this problem? The correct answer is still D, but I'm just curious
Math Expert
Joined: 02 Sep 2009
Posts: 58402
Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

### Show Tags

14 Dec 2013, 05:46
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.
_________________
Director
Joined: 14 Dec 2017
Posts: 516
Location: India
Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

### Show Tags

03 Aug 2018, 12:14
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
_________________
Non-Human User
Joined: 09 Sep 2013
Posts: 13247
Re: A rectangle is inscribed in a circle of diameter 2. Which of  [#permalink]

### Show Tags

25 Aug 2019, 11:52
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: A rectangle is inscribed in a circle of diameter 2. Which of   [#permalink] 25 Aug 2019, 11:52
Display posts from previous: Sort by