Question

One side of an equilateral triangle is 24 cm. The mid-points of its sides are joined to form another triangle is formed by joining the midpoints of this triangle and the process is continued indefinitely. Determine the sum of the perimeters of all such triangles.

Answer 1

According to given condition

sides of triangles will be $a,\frac{a}{2},\frac{a}{2^2},...........\infty$

So perimeters of the triangle will be $3a,\frac{3a}{2},\frac{3a}{2^2},...........\infty$

Hence sum $=3a(1+\frac{1}{2}+..........+\infty )=\frac{3a}{1-\frac{1}{2}}=6a=6\times 24cm=144cm$