SOLUTION: Mid-points of the side of an equilateral triangle of side 18 cm are joined to form another triangle, whose mid-points are further joined to form a different triangle and this p

Algebra ->  Triangles -> SOLUTION: Mid-points of the side of an equilateral triangle of side 18 cm are joined to form another triangle, whose mid-points are further joined to form a different triangle and this p      Log On


   

Question 876826: Mid-points of the side of an equilateral
triangle of side 18 cm are joined to form
another triangle, whose mid-points are
further joined to form a different triangle and
this process is repeated indefinitely. The sum
of the perimeters of all the triangles will be
Answer by fcabanski(1391) About Me  (Show Source):
You can put this solution on YOUR website!
The perimeter of the original triangle is 3*18=54cm. The triangle formed by the first midpoint connection has sides of 9cm. Its perimeter is 3*9cm = 27cm. The third triangle has sides of 4.5cm. Its perimeter is 3*4.5cm = 13.5cm.


The total perimeter is 3*(18+9+4.5+...)


18+9+4.5... is a geometric series 18/(2^n) where n is 0 to infinity.


The sum of an infinite geometric series xr^0+xr^1+xr^2+...xr^n+... converges to a certain value which is x/(1-r). In this case x=18 and r=1/2. So the sum is 18/(1/2) = 18*2 =36.


The total perimeter is 3*36 = 108 cm.