Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 01 Sep 2010
Posts: 20

Three identical circles are inscribed within an equilateral
[#permalink]
Show Tags
Updated on: 22 Nov 2013, 01:33
Question Stats:
70% (02:24) correct 30% (02:32) wrong based on 208 sessions
HideShow timer Statistics
Attachment:
Triangle.jpg [ 9.38 KiB  Viewed 30300 times ]
Three identical circles are inscribed within an equilateral triangle, as shown above. If each of the radii of the circles is 3, what is the perimeter of the triangle? A. 18(1+√3) B. 9(1+2√3) C. 18√3 D. 9(1+√3) E. 6(1+√3) I was able to solve it by guesstimating the answer, however I would love to see a detailed explanation one. (Question source  Master GMAT) Thanks.
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by eladshush on 28 Sep 2010, 06:40.
Last edited by Bunuel on 22 Nov 2013, 01:33, edited 1 time in total.
Renamed the topic and edited the question.




Math Expert
Joined: 02 Sep 2009
Posts: 49320

Re: Circles in a triangle
[#permalink]
Show Tags
28 Sep 2010, 07:33




Intern
Joined: 01 Sep 2010
Posts: 20

Re: Circles in a triangle
[#permalink]
Show Tags
28 Sep 2010, 08:15
Thank you Bunuel. This was very useful as always.



Manager
Joined: 27 Mar 2010
Posts: 98

Re: Circles in a triangle
[#permalink]
Show Tags
29 Sep 2010, 02:12
I guessed this by using an approximate value of each side of the triangle...If u see the figure then it can be visualized that each side of the equilateral triangle have a length slightly greater than the sum of the 4 radius=4*3 =12. Therefore perimeter must be greater than 12+12+12=36 approx...
which is choice A.



Intern
Joined: 18 Jul 2010
Posts: 45

Re: Circles in a triangle
[#permalink]
Show Tags
29 Sep 2010, 03:21
Just.... How do you prove that angle BAE (with E the center of the circle on the left hand side) is equal to 30 degree ???



Math Expert
Joined: 02 Sep 2009
Posts: 49320

Re: Circles in a triangle
[#permalink]
Show Tags
29 Sep 2010, 03:27
utin wrote: I guessed this by using an approximate value of each side of the triangle...If u see the figure then it can be visualized that each side of the equilateral triangle have a length slightly greater than the sum of the 4 radius=4*3 =12. Therefore perimeter must be greater than 12+12+12=36 approx...
which is choice A. This approach would give 100% accurate answer if only one option were more than 36, but in our case both A and B are more than 36. So chances are 50/50 to pick the right answer. Still good way of thinking for educated guess. alexn49 wrote: Just.... How do you prove that angle BAE (with E the center of the circle on the left hand side) is equal to 30 degree ??? The big triangle is equilateral, so the angels are 60 degrees each and BAE is half of it, so 30 degrees.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 18 Jul 2010
Posts: 45

Re: Circles in a triangle
[#permalink]
Show Tags
29 Sep 2010, 03:41
I reformulate : How do you know BAE is half ?



Math Expert
Joined: 02 Sep 2009
Posts: 49320

Re: Circles in a triangle
[#permalink]
Show Tags
29 Sep 2010, 04:00



Intern
Joined: 18 Jul 2010
Posts: 45

Re: Circles in a triangle
[#permalink]
Show Tags
29 Sep 2010, 04:42
Of course' ! Thx... Kudos !!!



Current Student
Joined: 06 Oct 2009
Posts: 90
Location: Mexico
Concentration: Entrepreneurship, Finance
GPA: 3.85
WE: Sales (Commercial Banking)

Re: Circles in a triangle
[#permalink]
Show Tags
30 Sep 2010, 11:41
I have a more straightforward way to solve it If we take the three centers of the inner circles we can draw an isoceles triangle with a total perimeter of 27 (9 X 3). There is a relationship between increase in in the side of an isoceles triangle and the perimeter of it. In that case we are increasing the above mentioned triangle in three. as it is the length of the ratio the inner circles. In that case we will increase the perimeter from 27 to (9 + 3 ) = 12 X 3 = 36. The only answer that has an aproximate value to 36 is A. Please comment.



Retired Moderator
Joined: 02 Sep 2010
Posts: 772
Location: London

Re: Circles in a triangle
[#permalink]
Show Tags
30 Sep 2010, 11:48
Bull78 wrote: I have a more straightforward way to solve it
If we take the three centers of the inner circles we can draw an isoceles triangle with a total perimeter of 27 (9 X 3).
There is a relationship between increase in in the side of an isoceles triangle and the perimeter of it. In that case we are increasing the above mentioned triangle in three. as it is the length of the ratio the inner circles. In that case we will increase the perimeter from 27 to (9 + 3 ) = 12 X 3 = 36. The only answer that has an aproximate value to 36 is A.
Please comment. I dont think I understand what you just did there. The triangle formed by joining the centers, is not isoceles, it is equilateral and also its perimeter is 6*3=18 and not 27. And finally not sure where you get that 36 from. The answers above are the exact answers not approximations
_________________
Math writeups 1) Algebra101 2) Sequences 3) Set combinatorics 4) 3D geometry
My GMAT story
GMAT Club Premium Membership  big benefits and savings



Math Expert
Joined: 02 Sep 2009
Posts: 49320

Re: Circles in a triangle
[#permalink]
Show Tags
30 Sep 2010, 12:01
shrouded1 wrote: Bull78 wrote: I have a more straightforward way to solve it
If we take the three centers of the inner circles we can draw an isoceles triangle with a total perimeter of 27 (9 X 3).
There is a relationship between increase in in the side of an isoceles triangle and the perimeter of it. In that case we are increasing the above mentioned triangle in three. as it is the length of the ratio the inner circles. In that case we will increase the perimeter from 27 to (9 + 3 ) = 12 X 3 = 36. The only answer that has an aproximate value to 36 is A.
Please comment. I dont think I understand what you just did there. The triangle formed by joining the centers, is not isoceles, it is equilateral and also its perimeter is 6*3=18 and not 27. And finally not sure where you get that 36 from. The answers above are the exact answers not approximations Also I'd add that the answer choices A and B are both more than 36: \(A\approx{49}\) and \(B\approx{40}\). As for 36, we can think in the following way: the side of the triangle must be more than \(4r=12\), so the perimeter must be more than \(3*12=36\) (see utin's post above). This approach would give 100% accurate answer if only one option were more than 36, but in our case both A and B are more than 36. So chances are 50/50 to pick the right answer. Still good way of thinking for educated guess. Hope it's clear.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Senior Manager
Joined: 06 Jun 2009
Posts: 299
Location: USA
WE 1: Engineering

Re: Circles in a triangle
[#permalink]
Show Tags
30 Sep 2010, 12:19
Makes sense. Educated guess can be really helpful if one has no idea of how to proceed with the problem. In this problem, by guessing one is looking for something which is 36+ A is about 49 B is about 40 Most of the ppl guessing this one will end up with B .......... it helps to know the correct way to solve question Thanks Bunuel.
_________________
All things are possible to those who believe.



SVP
Joined: 06 Sep 2013
Posts: 1803
Concentration: Finance

Re: Circles in a triangle
[#permalink]
Show Tags
21 Nov 2013, 14:59
eladshush wrote: HI all, Please consider the following problem: Attachment: Triangle.jpg Three identical circles are inscribed within an equilateral triangle, as shown above. If each of the radii of the circles is 3, what is the perimeter of the triangle? a. 18(1+√3) b. 9(1+2√3) c. 18√3 d. 9(1+√3) e. 6(1+√3) I was able to solve it by guesstimating the answer, however I would love to see a detailed explanation one. (Question source  Master GMAT) Thanks. Nice explanation Bunuel. Honestly, I could'nt find an easy way so I just ballparked. So we have that each side is more than 2(2r) >12 so three sides >36. The only answer choice that makes sense is A. Hope it helps Cheers! J



Manager
Joined: 18 Oct 2013
Posts: 74
Location: India
Concentration: Technology, Finance
Schools: Duke '16, Johnson '16, Kelley '16, Tepper '16, Marshall '16, McDonough '16, Insead '14, HKUST '16, HSG '15, Schulich '15, Erasmus '16, IE April'15, Neeley '15
GMAT 1: 580 Q48 V21 GMAT 2: 530 Q49 V13 GMAT 3: 590 Q49 V21
WE: Information Technology (Computer Software)

Re: Three identical circles are inscribed within an equilateral
[#permalink]
Show Tags
01 Mar 2014, 14:22
This question can be solved in 3 easy step 1. calculate the area of 3 inscribed circle 2. calculate the approximate side of equilateral triangle from answers and compute area of equilateral triangle 3. compare the answer correctly.
Solution Area of 3 inscribed circle = 3*3.14*9 =84. So area of triangle must be greater than 84 as circle are inscribed in triangle
a. 18(1+√3) =>6(2.7) =>16 A=(256*√3)/4=64√3 (approximately greater than 100 ) = 108 b. 9(1+2√3) =>3(4.46) => 13.3 =13 A=(169*√3)/4= 42√3(approximately 70) =74 c. 18√3 => 6√3 => 10.3 =10 . No need of calculation d. 9(1+√3) => 3(2.7) => 8 .No need of calculation e. 6(1+√3) => 2(2.7) => 6 .No need of calculation
SO answer is A.



SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1834
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)

Re: Three identical circles are inscribed within an equilateral
[#permalink]
Show Tags
03 Mar 2014, 01:20
vikrantgulia wrote: This question can be solved in 3 easy step 1. calculate the area of 3 inscribed circle 2. calculate the approximate side of equilateral triangle from answers and compute area of equilateral triangle 3. compare the answer correctly.
Solution Area of 3 inscribed circle = 3*3.14*9 =84. So area of triangle must be greater than 84 as circle are inscribed in triangle
a. 18(1+√3) =>6(2.7) =>16 A=(256*√3)/4=64√3 (approximately greater than 100 ) = 108 b. 9(1+2√3) =>3(4.46) => 13.3 =13 A=(169*√3)/4= 42√3(approximately 70) =74 c. 18√3 => 6√3 => 10.3 =10 . No need of calculation d. 9(1+√3) => 3(2.7) => 8 .No need of calculation e. 6(1+√3) => 2(2.7) => 6 .No need of calculation
SO answer is A. They are asking perimeter of the big triangle. Would this method still hold true. ?
_________________
Kindly press "+1 Kudos" to appreciate



Retired Moderator
Joined: 20 Dec 2013
Posts: 178
Location: United States (NY)
GMAT 1: 640 Q44 V34 GMAT 2: 710 Q48 V40 GMAT 3: 720 Q49 V40
GPA: 3.16
WE: Consulting (Venture Capital)

Re: Three identical circles are inscribed within an equilateral
[#permalink]
Show Tags
02 Jul 2014, 16:48
great explanation as always, thanks Bunuel
_________________
MY GMAT BLOG  ADVICE  OPINIONS  ANALYSIS



Board of Directors
Joined: 17 Jul 2014
Posts: 2683
Location: United States (IL)
Concentration: Finance, Economics
GPA: 3.92
WE: General Management (Transportation)

Re: Three identical circles are inscribed within an equilateral
[#permalink]
Show Tags
26 Oct 2015, 15:38
picked B, but purely guess. I could eliminate C, D, and E. Suppose we draw a diagonal for each circle so that to form 3 equilateral triangles with sides 6 now we have 6 equal sides of these small equilateral triangles, which is 36. But it has to be slightly more than 36, since we have 3 equal parts that are not covered between these 3 small triangles.
C,D,and E are <36. Between A and B...picked B.
bunnuel's explanation is good...have to analyze it more...



Board of Directors
Joined: 17 Jul 2014
Posts: 2683
Location: United States (IL)
Concentration: Finance, Economics
GPA: 3.92
WE: General Management (Transportation)

Re: Three identical circles are inscribed within an equilateral
[#permalink]
Show Tags
03 Jan 2016, 12:52
bumped into it another time...still had difficulties in understanding the process to solve it...nevertheless, I could eliminate C, D, and E, and after some more thinking, B as well..
so we have 3 circles, diameter of each is 6. thus, the length of each side should be more than 12. now we have 12+ * 3 = 36+.
A. 18(1+√3) = 18+18√3 = 18+slightly less than 36, or overall slightly less than 54. of these two, picked A. B. 9(1+2√3) = 9+ 18√3, so 9+slightly less than 36, or overall > slightly less than 45. C. 18√3 = less than 36, so out. D. 9(1+√3) = 9+9√3 = 8+18 => slightly less than 26, so out. E. 6(1+√3) = 6 + 6√3 = that's slightly less than 18, so out.



NonHuman User
Joined: 09 Sep 2013
Posts: 8159

Re: Three identical circles are inscribed within an equilateral
[#permalink]
Show Tags
12 Jun 2018, 08:10
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.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources




Re: Three identical circles are inscribed within an equilateral &nbs
[#permalink]
12 Jun 2018, 08:10






