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

It is currently 15 Aug 2018, 11:53

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

In the following figure, AB is the diameter of semicircle with center

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

Hide Tags

Manager
Manager
User avatar
P
Joined: 22 Feb 2018
Posts: 246
In the following figure, AB is the diameter of semicircle with center  [#permalink]

Show Tags

New post 12 Jul 2018, 22:10
2
3
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

50% (02:44) correct 50% (01:17) wrong based on 10 sessions

HideShow timer Statistics

In the following figure, AB is the diameter of semicircle with center M. Two semi-circles are drawn with AM and MB as the diameters. A circle is drawn such that it is tangent to all the three semi-circles. If AM = \(r\) units, find the radius of the circle (in terms of \(r\)).
Image


A) \(\frac{r}{2}\)

B) \(\frac{r}{3}\)

C) \(\frac{r}{4}\)

D) \(\frac{r}{{\sqrt{2}}}\)

E) \(\frac{2r}{5}\)


Attachment:
semi.PNG
semi.PNG [ 16.86 KiB | Viewed 299 times ]

_________________

Good, good Let the kudos flow through you

Intern
Intern
User avatar
S
Joined: 26 Jul 2017
Posts: 35
Location: India
GMAT 1: 570 Q48 V20
WE: Information Technology (Computer Software)
GMAT ToolKit User
In the following figure, AB is the diameter of semicircle with center  [#permalink]

Show Tags

New post Updated on: 12 Jul 2018, 23:36
Princ wrote:
In the following figure, AB is the diameter of semicircle with center M. Two semi-circles are drawn with AM and MB as the diameters. A circle is drawn such that it is tangent to all the three semi-circles. If AM = \(r\) units, find the radius of the circle (in terms of \(r\)).
Attachment:
The attachment semi.PNG is no longer available


A) \(\frac{r}{2}\)
B) \(\frac{r}{3}\)
C) \(\frac{r}{4}\)
D) \(\frac{r}{{\sqrt{2}}}\)
E) \(\frac{2r}{5}\)


As per the figure, the 3 centers of the 3 inscribed circle form an isosceles trianges with sides 0.5r+r1, 0.5r+r1,r. find the length of altitude( which bisects the side with length r).

triangle FCD is right angled triangle

Or \((CD)^2 = (CG)^2 + (DG)^2\)
Or \((Alt)^2 = (r)^2 + (0.5r + r1)^2\)
Or \((Alt) = \sqrt{r1(r+r1)}\)

Make a equation for the outermost circle radius as
\(r= r1 + \sqrt{r(r+r1)}\)
Or \((r-r1)^2 = r(r+r1)\)
Or \(3*r*r1=r^2\)
Or \(r1 = (r/3)\)
Hence Option B

Cheers :)
_________________
Please hit kudos, As it will encourage me to be more active at the forum.
The fact that you aren't where you want to be, should be enough motivation.
Attachments

Pun91 Geometry.png
Pun91 Geometry.png [ 260.18 KiB | Viewed 235 times ]


_________________

Please hit kudos, if you like this post
The fact that you aren't where you want to be, should be enough motivation.


Originally posted by pun91 on 12 Jul 2018, 22:57.
Last edited by pun91 on 12 Jul 2018, 23:36, edited 1 time in total.
Senior Manager
Senior Manager
User avatar
G
Status: Asst. Manager
Joined: 01 Oct 2017
Posts: 473
Location: India
Concentration: Operations, Entrepreneurship
GPA: 4
WE: Supply Chain Management (Energy and Utilities)
Premium Member CAT Tests
Re: In the following figure, AB is the diameter of semicircle with center  [#permalink]

Show Tags

New post 12 Jul 2018, 23:08
Princ wrote:
In the following figure, AB is the diameter of semicircle with center M. Two semi-circles are drawn with AM and MB as the diameters. A circle is drawn such that it is tangent to all the three semi-circles. If AM = \(r\) units, find the radius of the circle (in terms of \(r\)).
Attachment:
The attachment semi.PNG is no longer available


A) \(\frac{r}{2}\)
B) \(\frac{r}{3}\)
C) \(\frac{r}{4}\)
D) \(\frac{r}{{\sqrt{2}}}\)
E) \(\frac{2r}{5}\)


Please refer the affixed diagram,

OMD is a right angled triangle,
So, \(OD^2=OM^2+MD^2\)
Or, (\((r_{1}+(\frac{r}{2}))^2\)=\((r-r_{1})^2+(\frac{r}{2})^2\)
Or, \(rr_{1}=r^2-2rr_{1}\)
Or, \(r^2=3rr_{1}\)
Or, \(r=3r_{1}\)
Or, \(r_{1}=\frac{r}{3}\)

Ans. (B)
Attachments

Semicircle.JPG
Semicircle.JPG [ 22.36 KiB | Viewed 221 times ]


_________________

Regards,

PKN

Rise above the storm, you will find the sunshine

Intern
Intern
avatar
Joined: 14 Feb 2017
Posts: 1
Re: In the following figure, AB is the diameter of semicircle with center  [#permalink]

Show Tags

New post 12 Jul 2018, 23:43
Thanks! This shows that even the seemingly complex questions in GMAT can be solved by simple and effective techniques like that of Pythagorus Theorem.

pun91 wrote:
Princ wrote:
In the following figure, AB is the diameter of semicircle with center M. Two semi-circles are drawn with AM and MB as the diameters. A circle is drawn such that it is tangent to all the three semi-circles. If AM = \(r\) units, find the radius of the circle (in terms of \(r\)).
Attachment:
semi.PNG


A) \(\frac{r}{2}\)
B) \(\frac{r}{3}\)
C) \(\frac{r}{4}\)
D) \(\frac{r}{{\sqrt{2}}}\)
E) \(\frac{2r}{5}\)


As per the figure, the 3 centers of the 3 inscribed circle form an isosceles trianges with sides 0.5r+r1, 0.5r+r1,r. find the length of altitude( which bisects the side with length r).

triangle FCD is right angled triangle

Or \((CD)^2 = (CG)^2 + (DG)^2\)
Or \((Alt)^2 = (r)^2 + (0.5r + r1)^2\)
Or \((Alt) = \sqrt{r1(r+r1)}\)

Make a equation for the outermost circle radius as
\(r= r1 + \sqrt{r(r+r1)}\)
Or \((r-r1)^2 = r(r+r1)\)
Or \(3*r*r1=r^2\)
Or \(r1 = (r/3)\)
Hence Option B

Cheers :)
_________________
Please hit kudos, As it will encourage me to be more active at the forum.
The fact that you aren't where you want to be, should be enough motivation.
Re: In the following figure, AB is the diameter of semicircle with center &nbs [#permalink] 12 Jul 2018, 23:43
Display posts from previous: Sort by

In the following figure, AB is the diameter of semicircle with center

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

Events & Promotions

PREV
NEXT


GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

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

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.