Question

An equilateral triangle is drawn by joining the midpoints of the sides of another equilateral triangle. A third equilateral triangle is drawn inside the second one joining the midpoints of the sides of the second equilateral triangle, and the process continues infinitely. Find the sum of the areas of all the equilateral triangles, if the side of the largest equilateral triangle is 8 units.

(a) 32√3 units 

(b) 64√3 units

(c) 64 units

(d) 64/√3 units

Answer 1

Correct option (d) 64√3 units

Explanation:

The side of the first equilateral triangle being 8 units, the first area is 16√3 square units. The second area would be 1/4 of area of largest triangle and so on. 16√3,4√3,√3/4,√3/16,.....

The infinite sum of this series — 16√3 /(1 − 1/4) =64√3 square units.