BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

points are turn joined

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
User avatar
GMAT Instructor
Posts: 3650
Joined: Wed Jan 21, 2009 4:27 am
Location: India
Thanked: 267 times
Followed by:80 members
GMAT Score:760

points are turn joined

by sanju09 » Sat Jun 26, 2010 1:35 am
The side of an equilateral triangle measures 96. The mid points of its side are joined to form another triangle whose mid points are turn joined to form still another triangle. This process is repeated indefinitely. What is the sum of the perimeter of all the triangles so formed along with the original triangle?
(A) 576
(B) 1152
(C) 1176
(D) 1275
(E) 2304
The mind is everything. What you think you become. -Lord Buddha



Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001

www.manyagroup.com
Top
Source: — Problem Solving |
User avatar
Legendary Member
Posts: 1460
Joined: Tue Dec 29, 2009 1:28 am
Thanked: 135 times
Followed by:7 members

by selango » Sat Jun 26, 2010 2:32 am
First triangle sides are 96

second triangle sides will be 48

third triangle sides will be 24 and so on

Sum perimeters of all triangles =3*96+3*48+3*24+3*12.......

=3*96[1+1/2+1/4+1/8+1/16....]


-->288[1+1/2+1/4+1/8+1/16....]=x

-->[1+1/2+1/4+1/8+1/16....] =x/288

sigma[(1/2)^n]=x/288

sub ans choices in x.It must satisfy left hand side equation.

I cant proceed from this.Any suggestions.
Top
User avatar
Master | Next Rank: 500 Posts
Posts: 392
Joined: Sun May 16, 2010 2:42 am
Location: Bangalore, India
Thanked: 116 times
Followed by:10 members
GMAT Score:770

by albatross86 » Sat Jun 26, 2010 2:36 am
The sides of the first inner triangle will be HALF of the side of the original triangle. The next one will be HALF of the previous one, and so on. Why? Equilateral triangles are symmetric and when you draw such a triangle out you will see this.

So in essence this is an infinite series of the form:

3*(96 + 96/2 + 96/4 + ...)

The sum of an infinite Geometric progression is a/ (1-r) where a is the first term and 2 is the common ratio

=> 3 * 96 / (1 - 0.5)

= 3 * 96 * 2

= 576

Pick A


NOTE: I've attached a picture I drew very amateurishly :D

In that you can see that angle ADE must be equal to AED so, Triangle ADE is equilateral too. Thus x = 48. If you do the same thing for the other two triangles BDF and FEC, you find that triangle DEF is an equilateral triangle with side 96/2 = 48.
Attachments
Equilateral.PNG
Top
User avatar
Master | Next Rank: 500 Posts
Posts: 392
Joined: Sun May 16, 2010 2:42 am
Location: Bangalore, India
Thanked: 116 times
Followed by:10 members
GMAT Score:770

by albatross86 » Sat Jun 26, 2010 2:43 am
If you do not like remembering formulae and prefer artistic/ intuitive / elegant approaches to problems:

First reach the point:

Sum = 3*96* ( 1 + 1/2 + 1/4 + 1/8 + .... )

Consider this sum: 1/2 + 1/4 + 1/8 + ....

If I give you half a pizza, and you're still hungry - you want moaarr! I give you half of the half I have. But you're not satisfied and keep asking for more. I'm going to keep giving you half of what I hold in my hands.

If I do this INFINITELY, the amount of pizza you hold will TEND towards one entire pizza, since I cannot give you more than 1 pizza.

The only way to grasp the abstract concept of infinity is to imagine that at that point, this impossibility of me continually giving you half of what I hold, at infinity, would result in me giving you the entire pizza. Ofcourse in reality this would never happen because infinity cannot be experienced, atleast on this realm :D

So a series of the form (1/2 + 1/4 + 1/8 + ...) is going to tend towards equalling 1, and thus an infinite series will be equal to 1

SO we have 3*96*(1+1) = 576
Top
User avatar
Legendary Member
Posts: 1460
Joined: Tue Dec 29, 2009 1:28 am
Thanked: 135 times
Followed by:7 members

by selango » Sat Jun 26, 2010 2:52 am
albatross86,

wow..Nice explanation.

So the sum of an infinite Geometric progression is a/ (1-r) where a is the first term and r is the common ratio

-->288[1/1-0.5]=288*2=576
Top
Post new topic Post Reply
• Page 1 of 1