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

 It is currently 20 Jan 2019, 01:04

### 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

## Events & Promotions

###### Events & Promotions in January
PrevNext
SuMoTuWeThFrSa
303112345
6789101112
13141516171819
20212223242526
272829303112
Open Detailed Calendar
• ### FREE Quant Workshop by e-GMAT!

January 20, 2019

January 20, 2019

07:00 AM PST

07:00 AM PST

Get personalized insights on how to achieve your Target Quant Score.
• ### GMAT Club Tests are Free & Open for Martin Luther King Jr.'s Birthday!

January 21, 2019

January 21, 2019

10:00 PM PST

11:00 PM PST

Mark your calendars - All GMAT Club Tests are free and open January 21st for celebrate Martin Luther King Jr.'s Birthday.

# ABCD is a rectangle inscribed in a circle. If the length of AB is thr

Author Message
TAGS:

### Hide Tags

Intern
Joined: 18 Sep 2010
Posts: 45
ABCD is a rectangle inscribed in a circle. If the length of AB is thr  [#permalink]

### Show Tags

16 Apr 2011, 23:04
14
00:00

Difficulty:

85% (hard)

Question Stats:

53% (02:15) correct 47% (02:48) wrong based on 151 sessions

### HideShow timer Statistics

ABCD is a rectangle inscribed in a circle.

Attachments

geo.png [ 24.52 KiB | Viewed 4687 times ]

_________________

(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Senior Manager
Joined: 08 Nov 2010
Posts: 336
Re: ABCD is a rectangle inscribed in a circle. If the length of AB is thr  [#permalink]

### Show Tags

19 Apr 2011, 04:47
7
1
i did it a bit different.

Max size for a rectangle is a square.
and minimum is when the width or length = to 0 (or almost 0)

so lets check the max and than we know that everything below the max is a possibility.

so if we know the radii is 1 - we know that if it was a square its diagonals will be 2 each.
so the area of the rectangle will be < 2*2/2 = 2

so - I, II is below 2, above 0 - than - they are ok.
_________________
##### General Discussion
Retired Moderator
Joined: 20 Dec 2010
Posts: 1809
Re: ABCD is a rectangle inscribed in a circle. If the length of AB is thr  [#permalink]

### Show Tags

16 Apr 2011, 23:36
4
annmary wrote:
hello
i cant solve this geometry question , plz help!
tanx

Sol:
Let l=length
w=wdith
d=diagonal=2
$$Area=l*w=l*\sqrt{d^2-l^2}$$

$$l*\sqrt{2^2-l^2}$$

$$l*\sqrt{4-l^2}=Area$$

Squaring both sides;
$$l^2*(4-l^2)=(Area)^2$$
$$4l^2-l^4=(Area)^2$$
$$-l^4+4l^2-(Area)^2=0$$

$$Let \hspace{2} l^2=x$$

$$-x^2+4x-(Area)^2=0$$

$$D=\sqrt{b^2-4*a*c}$$

$$D=\sqrt{4^2-4*(-1)*-(Area)^2}$$

$$D=\sqrt{16-4*(Area)^2}$$

In inside of the root must be greater or equal to 0 to provide valid roots of the equation.
$$16-4*(Area)^2 \ge 0$$

$$4*(Area)^2 \le 16$$

$$(Area)^2 \le 4$$

$$(Area) \le 2$$

Only I and II are less than 2.

Ans: "D"
_________________
Intern
Joined: 18 Sep 2010
Posts: 45
Re: ABCD is a rectangle inscribed in a circle. If the length of AB is thr  [#permalink]

### Show Tags

17 Apr 2011, 05:12
thank you fluke
+5 kudos , but i just can give you +1 kudos
other 4 kudos :+1,+1,+1,+1
_________________

(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Manager
Joined: 17 Jan 2011
Posts: 220
Re: ABCD is a rectangle inscribed in a circle. If the length of AB is thr  [#permalink]

### Show Tags

17 Apr 2011, 13:20
1
1
Area cannot be 3.2 because area of circle is pi r^2 i.e. 3.14, so rule out III
Even if the length is very close to the diameter i.e. 1 approx, width has to be 1/100 to make the area .01 - possible
similarly 1.9 is also possible.
=>D
_________________

Good Luck!!!

***Help and be helped!!!****

Senior Manager
Joined: 08 Nov 2010
Posts: 336
Re: ABCD is a rectangle inscribed in a circle. If the length of AB is thr  [#permalink]

### Show Tags

19 Apr 2011, 05:07
thanks old friend.
_________________
Manager
Joined: 09 Jan 2010
Posts: 82
Re: ABCD is a rectangle inscribed in a circle. If the length of AB is thr  [#permalink]

### Show Tags

01 May 2011, 19:10
1
area of rectangle = l*b

diagonal is 2
also d (diagonal) = sqrt (l^2 + b^2)
l*b = l * sqrt (2 -l^2)
area is positive
therefore (2-l^2) >0
i.e sqrt2>l
similarly sqrt2>b
therefore area = l.b <sqrt 2.sqrt2 = 1.4 *1.4 = 1.96
therefore D is correct.
VP
Status: There is always something new !!
Affiliations: PMI,QAI Global,eXampleCG
Joined: 08 May 2009
Posts: 1015
Re: ABCD is a rectangle inscribed in a circle. If the length of AB is thr  [#permalink]

### Show Tags

01 May 2011, 21:29
1
Max area of the rectangle possible when diagonal = diameter = 2
Hence, l^2 + b^2 = 4. Implies l and b < 1.5 each.
Also, area of circle = 3.14 * 1^2 = 3.14. hence options C and E POE.
Now,
0<l <1.5 and 0<b<1.5. Thus area can be 0.01 and 1.9 both.
Hence D.
Intern
Joined: 03 Jul 2015
Posts: 10
Location: India
Re: ABCD is a rectangle inscribed in a circle. If the length of AB is thr  [#permalink]

### Show Tags

18 Jul 2015, 23:34
4
annmary wrote:
ABCD is a rectangle inscribed in a circle.

Rectangle $$ABCD$$ can be broken down into 2 congruent triangles, $$\triangle$$$$ABC$$ & $$\triangle$$$$CDA$$.

area of rectangle $$ABCD$$ = 2 * area of $$\triangle$$$$ABC$$

Let's draw a perpendicular from B to AC and denote the length by h.

Also, $$AC = 2r = 2$$

Now, area of $$\triangle$$$$ABC$$ $$= (1/2)*(2)*(h) = h$$

area of rectangle $$ABCD = 2h$$

Now maximum value of h can be r and minimum can be tending towards 0.

So, area of rectangle will vary between $$0$$ and $$2$$.

$$Answer D$$
Manager
Joined: 03 May 2013
Posts: 66
Re: ABCD is a rectangle inscribed in a circle. If the length of AB is thr  [#permalink]

### Show Tags

16 Oct 2015, 23:27
15 seconds solution
the max area of a rectangle could be the area of squire that can be inscribed in circle with vertices on edges of circle , hence max area could be 2*2 / 2 as 2 is the max length of dignol of squire, hence D
Intern
Joined: 05 Dec 2016
Posts: 7
Re: ABCD is a rectangle inscribed in a circle. If the length of AB is thr  [#permalink]

### Show Tags

05 May 2017, 06:26
Trigonometric Approach

Since the radius is equal to 1, the diagonal of rectangle is equal to 2r = 2

Hence CD= 2 sin x and AD= 2 cos x

Area of the rectangle ABCD= AD.CD = (2sinx)(2cosx)= 2(2sinx.cosx) = 2 sin 2x

Now sin2x has maximum value 1 & minimum value 0

Hence 0 < area (ABCD) < 2

Therefore OPTION D
Non-Human User
Joined: 09 Sep 2013
Posts: 9451
Re: ABCD is a rectangle inscribed in a circle. If the length of AB is thr  [#permalink]

### Show Tags

08 Jun 2018, 03:54
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: ABCD is a rectangle inscribed in a circle. If the length of AB is thr &nbs [#permalink] 08 Jun 2018, 03:54
Display posts from previous: Sort by