Question

One side of an equilateral triangle is 24 cm. The mid points of its sides are joined to form another triangle whose mid points are in term joined to form still another triangle. This process continues indefinitely. Find the sum of the perimetres (in cm) of all such triangles.

Answer 1

The perimeter of the biggest triangle we have is $24\times 3$ i.e. $72.$
The next smaller triangle has side half of the biggest triangle and so its perimeter will be $36.$
This goes on similarly and we have the formation of an infinite G.P. series
The sum of the series is given by $\frac{a}{1-r}=\frac{72}{1-0.5}=144cm.$