It is currently 18 Mar 2018, 14:33

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 June
Open Detailed Calendar

The figure above shows an equilateral triangle with three circles. Eac

Author Message
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 44298
The figure above shows an equilateral triangle with three circles. Eac [#permalink]

Show Tags

08 Mar 2018, 21:26
Expert's post
5
This post was
BOOKMARKED
00:00

Difficulty:

(N/A)

Question Stats:

63% (02:41) correct 37% (02:19) wrong based on 30 sessions

HideShow timer Statistics

The figure above shows an equilateral triangle with three circles. Each point of contact between two circles or between a circle and the triangle is a point of tangency. If the triangle has a height of 18, what is the combined area of the two smaller circles?

A. 4π
B. 8π
C. 16π
D. 36π
E. 44π

[Reveal] Spoiler:
Attachment:

Equilateral_triangle_circle_packing_1 (1).png [ 13.97 KiB | Viewed 608 times ]
[Reveal] Spoiler: OA

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 44298
Re: The figure above shows an equilateral triangle with three circles. Eac [#permalink]

Show Tags

08 Mar 2018, 21:27
Expert's post
1
This post was
BOOKMARKED
Bunuel wrote:

The figure above shows an equilateral triangle with three circles. Each point of contact between two circles or between a circle and the triangle is a point of tangency. If the triangle has a height of 18, what is the combined area of the two smaller circles?

A. 4π
B. 8π
C. 16π
D. 36π
E. 44π

[Reveal] Spoiler:
Attachment:
Equilateral_triangle_circle_packing_1 (1).png

For other subjects:
ALL YOU NEED FOR QUANT ! ! !
_________________
SVP
Joined: 08 Jul 2010
Posts: 2017
Location: India
GMAT: INSIGHT
WE: Education (Education)
The figure above shows an equilateral triangle with three circles. Eac [#permalink]

Show Tags

09 Mar 2018, 00:31
1
KUDOS
Expert's post
2
This post was
BOOKMARKED
Bunuel wrote:

The figure above shows an equilateral triangle with three circles. Each point of contact between two circles or between a circle and the triangle is a point of tangency. If the triangle has a height of 18, what is the combined area of the two smaller circles?

A. 4π
B. 8π
C. 16π
D. 36π
E. 44π

[Reveal] Spoiler:
Attachment:
The attachment Equilateral_triangle_circle_packing_1 (1).png is no longer available

The radius of an In-Circle in an equilateral triangle = (1/3)* Height of equilateral triangle(DERIVATION IS ATTACHED HERE)

therefore Radius of the bigger circle = (1/3)*18 = 6

Now The Height of Triangle ADE = 18-(2*radius of bigger circle) = 18 - 12 = 6

Now the radius of the smaller circle = (1/3)* Height of equilateral triangle ADE = (1/3)*6 = 2

i.e. Area of Both the smaller circle = 2* πr^2 = 2*π*2^2 = 8π

Attachments

File comment: www.GMATinsight.com

Screen Shot 2018-03-09 at 1.08.46 PM.png [ 496.21 KiB | Viewed 329 times ]

_________________

Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION

examPAL Representative
Joined: 07 Dec 2017
Posts: 210
Re: The figure above shows an equilateral triangle with three circles. Eac [#permalink]

Show Tags

09 Mar 2018, 02:11
Bunuel wrote:
The figure above shows an equilateral triangle with three circles. Each point of contact between two circles or between a circle and the triangle is a point of tangency. If the triangle has a height of 18, what is the combined area of the two smaller circles?

A. 4π
B. 8π
C. 16π
D. 36π
E. 44π

Since there is only one way to draw an equilateral triangle with an inscribed circle, we can trust the drawing.
That is, we'll visually estimate the answer and look for the closest answer choice.
This is an Alternative approach.

The height of the triangle is 18 so its base is 18$$\sqrt{2}$$ (because this is a 30-60-90 triangle) which is about 20*1.4 = 28
Then the area is about 18*28/2 = 18*14 = 140+80+32=252
Dividing our equilateral triangle into 9 identical smaller triangles, we can SEE that the two small circles are about half the area of 2 of the small triangles.
That is, about (1/2)*(2/9) of the total area or 252/9 which is a bit more than 25, say 30.
Option (B) is the only relevant choice.

** Note that with regular polygons and area-related questions it is almost always easier to estimate the answer instead of looking for an extremely complicated Precise geometrical solution.
Attachments

equi.png [ 38.58 KiB | Viewed 294 times ]

_________________

David
Senior tutor at examPAL
Signup for a free GMAT course

We won some awards:

Save up to \$250 on examPAL packages (special for GMAT Club members)

Manager
Joined: 04 Apr 2015
Posts: 103
Re: The figure above shows an equilateral triangle with three circles. Eac [#permalink]

Show Tags

09 Mar 2018, 11:19
GMATinsight wrote:
Bunuel wrote:

The figure above shows an equilateral triangle with three circles. Each point of contact between two circles or between a circle and the triangle is a point of tangency. If the triangle has a height of 18, what is the combined area of the two smaller circles?

A. 4π
B. 8π
C. 16π
D. 36π
E. 44π

[Reveal] Spoiler:
Attachment:
Equilateral_triangle_circle_packing_1 (1).png

The radius of an In-Circle in an equilateral triangle = (1/3)* Height of equilateral triangle(DERIVATION IS ATTACHED HERE)

therefore Radius of the bigger circle = (1/3)*18 = 6

Now The Height of Triangle ADE = 18-(2*radius of bigger circle) = 18 - 12 = 6

Now the radius of the smaller circle = (1/3)* Height of equilateral triangle ADE = (1/3)*6 = 2

i.e. Area of Both the smaller circle = 2* πr^2 = 2*π*2^2 = 8π

Radius =1/3 of the height is obvious since circumcenter,incenter of equilateral triangle lie on the same point. Since circumcenter divides the triangle in 2:1 ratio, the radius should be 1/3 of the height
Re: The figure above shows an equilateral triangle with three circles. Eac   [#permalink] 09 Mar 2018, 11:19
Display posts from previous: Sort by