Aside of an equilateral triangle is 10 cm. the midpoints of its sides are joined to form an inscribed equilateral triangle, and this process is continued. find the sum of the areas of the triangles of the process is continued without end.

Equilateral triangle is a triangle in which all its sides are equal.

Given:
side = 10 cm

Area of an equilateral triangle = √3/4  a²

1st: a = 10 ; A = 43.3 cm²
2nd: a = 5 ; A = 10.83 cm²
3rd: a = 2.5 ; A = 2.71 cm²
4th: a = 1.25 ; A = 0.68 cm²
5th: a = 0.625 ; A = 0.17 cm²
6th: a = 0.3125; A = 0.042 cm²
7th: a = 0.15625; A = 0.011.cm²

Summation: a1/1 -r ; a1 = 43.30 ; r = 0.25

43.30 / 1 - 0.25 = 43.30/0.75 = 57.73 cm²

The sum of the areas of the triangles if the process is continued without end is 57.73 cm²