points are turn joined

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.

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

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.

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

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

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