Infinite Triangles
The middle points of the sides of the triangle are joined formng a second triangle. Again a third triangle is formed by joining the middle points of this second triangle and this process is repeated infinitely. If the perimeter of outer triangle is P , then what will be the sum of perimeters of triangles thus formed ???
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
I think it is missing a key piece of information. The triangles are equilateral. If it is true for any triangle, then I can't prove it. But for equilateral triangles, it is easy. The sides of every next internal triangle inside its parent external triangle are half of the parent's sides, and hence, so is the perimeter. Therefore the sum of all the perimeters would be:
P+(1/2)P+(1/4)P+(1/8)P+ ....
or P[ 1+(1/2)+(1/4)+(1/8)+...]
which is an infinite geometric series with r=1/2. The sum of such series starting from 1 is 1/(1-r) which gives 1/(1-(1/2)) = 2.
Therefore the sum of all the perimeters would 2P.