Question

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

Answer 1

We know that the line joining the mid-points of the sides of a triangle is half the opposite side. The triangle being equilateral.
Sides are $a,\frac{a}{2},\frac{a}{4},\frac{a}{8},...G.P.$
Perimeters are $3a,\frac{3a}{2},\frac{3a}{4}......$
Sum of perimeters $=\frac{3a}{1-\frac{1}{2}}=6a=6\times 24=144$.