Last visit was: 20 Sep 2026, 08:31 It is currently 20 Sep 2026, 08:31
Home
Messages
Tests
Schools
Events
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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
805+ (Hard)|   Geometry|      
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 20 Sep 2026
Posts: 113,688
Own Kudos:
842,493
 [12]
Given Kudos: 111,672
Products:
GMAT Club Tests
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 113,688
Kudos: 842,493
 [12]
In the diagram, four circles of radius 1 with centers P, Q, R and S ar 10 Nov 2020, 03:30
Kudos
Add Kudos
11
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

95% (hard)

Question Stats:

32% (02:32) correct 68% (02:47) wrong based on 53 sessions

History

Date
Time
Result
Not Attempted Yet

In the diagram, four circles of radius 1 with centers P, Q, R and S are tangent to one another and sides of \(\triangle{ABC}\), as shown. Find the perimeter of \(\triangle{ABC}\).

(A) \(12+4\sqrt{2}\)

(B) \(12+4\sqrt{3}\)

(C) \(12+6\sqrt{2}\)

(D) \(12+6\sqrt{3}\)

(E) \(24\)

Show Spoiler
Attachment:
4circle-triangle 1.png
4circle-triangle 1.png [ 4.1 KiB | Viewed 12356 times ]
Attachment:
1.jpg
1.jpg [ 18.67 KiB | Viewed 10482 times ]


Are You Up For the Challenge: 700 Level Questions
ShowHide Answer
Official Answer
D
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 20 Sep 2026
Posts: 11,310
Own Kudos:
46,224
 [1]
Given Kudos: 339
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,310
Kudos: 46,224
 [1]
In the diagram, four circles of radius 1 with centers P, Q, R and S ar 10 Nov 2020, 23:31
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel

In the diagram, four circles of radius 1 with centers P, Q, R and S are tangent to one another and sides of \(\triangle{ABC}\), as shown. Find the perimeter of \(\triangle{ABC}\).

(A) \(12+4\sqrt{2}\)

(B) \(12+4\sqrt{3}\)

(C) \(12+6\sqrt{2}\)

(D) \(12+6\sqrt{3}\)

(E) \(24\)

Show more

Such questions almost always require some additional lines to be made.

STEP 1
Join P, R and S to get an equilateral triangle PRS.
So \(\angle PSR = 60\)

STEP 2
Join P, Q and S to get a \(\triangle PQS\).
\(\angle PSQ = 60\), and two sides are 2 and 4....2 and 4 come from the addition of radius 1.
Therefore \(\triangle PQS\) is a 30-60-90 triangle with sides \(2:2\sqrt{3}:4\)
Perimeter = \(2+2\sqrt{3}+4=6+2\sqrt{3}\)

STEP 3
Take \(\triangle ABC\) and \(\triangle PQS\)
All sides are parallel, so both are similar.
So the ratio of sides is same. Therefore, let us find the ratio between AC and PS.
\(\angle PSQ = \angle ACB=60\)

STEP 4
Join C, D and S to get aright angled triangle CDS.
Now, CS bisects the angle ACB, so \(\angle DCS=30\), and \(\triangle DCS\) is a 30-60-90 triangle with sides SD:DC:CS as \(1:\sqrt{3}:2\)
So AC=AD+DC=\(3+\sqrt{3}\)

STEP 5
The ratio of parallel sides.
\(PS:AC=2:3+\sqrt{3}\)
So ratio of perimeters of triangle \(PQS:ABC=2:3+\sqrt{3}=6+2\sqrt{3}:ABC\)
Perimeter of ABC = \(\frac{(3+\sqrt{3})(6+2\sqrt{3})}{2}=(3+\sqrt{3})(3+\sqrt{3})=(3+\sqrt{3})^2\)
\(9+3+6\sqrt{3}=12+6\sqrt{3}\)

D
Attachments

4circle-triangle 1.png
4circle-triangle 1.png [ 12.82 KiB | Viewed 11021 times ]

User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 20 Sep 2026
Posts: 113,688
Own Kudos:
842,493
Given Kudos: 111,672
Products:
GMAT Club Tests
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 113,688
Kudos: 842,493
In the diagram, four circles of radius 1 with centers P, Q, R and S ar 12 Nov 2020, 01:02
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel

In the diagram, four circles of radius 1 with centers P, Q, R and S are tangent to one another and sides of \(\triangle{ABC}\), as shown. Find the perimeter of \(\triangle{ABC}\).

(A) \(12+4\sqrt{2}\)

(B) \(12+4\sqrt{3}\)

(C) \(12+6\sqrt{2}\)

(D) \(12+6\sqrt{3}\)

(E) \(24\)

Show Spoiler
Attachment:
4circle-triangle 1.png
Attachment:
1.jpg
Show more

Part of Are You Up For the Challenge: 700 Level Questions collection.
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 39,231
Own Kudos:
1,162
Posts: 39,231
Kudos: 1,162
In the diagram, four circles of radius 1 with centers P, Q, R and S ar 02 Nov 2025, 09:43
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.
Moderator:
Bunuel
Math Expert
113688 posts