Question

One side of an equilateral triangle is $24cm$. The mid-points of its sides are joined to form another triangle whose mid points are in turn joined to form still another triangle. This process continues indefinitely. Find the sum of the perimeters of all the triangles.

Answer 1

Side length of equilateral triangle wise decrease by half after every turn.

sum of perimeter $=3\times 24+3\times 12+3\times \mathrm{6...........}$

$=3\times 24(1+\frac{1}{2}+\frac{1}{4}+.......\infty )$

$=3\times 24\times \frac{1}{1-\frac{1}{2}}$

$=3\times 2\times 24$

$=144$