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

It is currently 16 Oct 2019, 11:14

Close

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
Your Progress

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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

What percent of the area of the rhombus ABCD is the area of the circle

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
GMATH Teacher
User avatar
P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
What percent of the area of the rhombus ABCD is the area of the circle  [#permalink]

Show Tags

New post 07 Mar 2019, 15:49
00:00
A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

14% (01:41) correct 86% (01:50) wrong based on 22 sessions

HideShow timer Statistics

GMATH practice exercise (Quant Class 19)

Image

What percent of the area of the rhombus ABCD is the area of the circle that is inscribed in it?

(1) Angle BCD is equal to 60 degrees
(2) AB = 4

_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Intern
Intern
avatar
B
Joined: 09 Nov 2014
Posts: 11
Re: What percent of the area of the rhombus ABCD is the area of the circle  [#permalink]

Show Tags

New post 07 Mar 2019, 20:01
Can someone plz explain how A is the answer?

Posted from my mobile device
Math Expert
avatar
V
Joined: 02 Aug 2009
Posts: 7959
Re: What percent of the area of the rhombus ABCD is the area of the circle  [#permalink]

Show Tags

New post 07 Mar 2019, 20:19
2
What percent of the area of the rhombus ABCD is the area of the circle that is inscribed in it?

(1) Angle BCD is equal to 60 degrees
Join the diagonals as shown.. Triangle COB is 30-60-90 and sides are in ratio 1:\(\sqrt{3}\):2...
so CO = \(a\sqrt{3}\), and BO = a, when side BC = 2a.... so AREA of rhombus = 2a*2\(a\sqrt{3}\)
NEXT take triangle COX, by taking x on tangent .. so again Triangle COX is 30-60-90 and sides are in ratio 1:\(\sqrt{3}\):2...
Now CO =2*common ratio =\(a\sqrt{3}\), so radius of circle = \(\frac{CO}{2}=a\sqrt{3}\)/2
Thus area of circle = \(\pi*(a\sqrt{3}/2)^2=\pi*3a^2/4\)..
Fraction = \(\frac{\pi*3a^2}{4*2a*2a*sqrt(3)}\)... a will get cancelled out and we can find the fraction
Sufficient

(2) AB = 4
We do not the angles, we cannot say anything..
If all angles are 90, then the fraction will be different and if 60 and 120 then different
Insufficient

A
Attachments

07-Mar19-7h.gif
07-Mar19-7h.gif [ 5.74 KiB | Viewed 393 times ]


_________________
GMATH Teacher
User avatar
P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
What percent of the area of the rhombus ABCD is the area of the circle  [#permalink]

Show Tags

New post 08 Mar 2019, 06:39
fskilnik wrote:
GMATH practice exercise (Quant Class 19)

Image

What percent of the area of the rhombus ABCD is the area of the circle that is inscribed in it?

(1) Angle BCD is equal to 60 degrees
(2) AB = 4

Image

\(? = {{{S_{{\rm{circle}}}}} \over {{S_{{\rm{rhombus}}}}}}\)

\(\left( 1 \right)\,\,\,\left\{ \matrix{
BCD = {60^ \circ } \hfill \cr
ABCD\,\,{\rm{rhombus}} \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\Delta \,CBD\,\,{\rm{and}}\,\,\Delta ABD\,\,{\rm{equilaterals}}\,\,\,\left( * \right)\,\,\,\,\,\)

\(?\,\,\mathop = \limits^{\left( * \right)} \,\,\frac{{{S_{{\text{circle}}}}}}{{2 \cdot {S_{\Delta {\text{ABD}}}}}}\,\,\mathop = \limits^{\left( * \right)} \,\,\frac{{\pi \cdot O{E^2}}}{{2 \cdot \left( {\frac{1}{2} \cdot BD \cdot \frac{{BD \cdot \sqrt 3 }}{2}} \right)}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,? = {\left( {\frac{{OE}}{{BD}}} \right)^2}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\boxed{\,? = \frac{{OE}}{{BD}}\,}\)

\(\left\{ \matrix{
\,\Delta OEB\,\,\,\,\left[ {{{30}^ \circ },{{60}^ \circ },{{90}^ \circ }} \right] \hfill \cr
\,{1 \over 2}BD = OB\,\,\,{\rm{hyp}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left[ {{{30}^ \circ },{{60}^ \circ },{{90}^ \circ }} \right]} \,\,\,\,OE = {{OB \cdot \sqrt 3 } \over 2} \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {{OE} \over {BD}} = {{OE} \over {2 \cdot OB}} = {1 \over 2}\left( {{{\sqrt 3 } \over 2}} \right)\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\)


\(\left( 2 \right)\,\,{\rm{Insuff}}.\,\,\,\,\left( {{\rm{geometric}}\,\,{\rm{bifurcation}}\,{\rm{,}}\,\,{\rm{see}}\,\,{\rm{images}}} \right)\)


The correct answer is (A).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
GMAT Club Bot
What percent of the area of the rhombus ABCD is the area of the circle   [#permalink] 08 Mar 2019, 06:39
Display posts from previous: Sort by

What percent of the area of the rhombus ABCD is the area of the circle

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne