Question

Given an equilateral triangle T1 with side 24 cm, a second triangle T2 is formed by joining the midpoints of the sides of T1. Then a third triangle T3 is formed by joining the midpoints of the sides of T2. If this process of forming triangles is continued, the sum of the areas, in sq cm, of infinitely many such triangles T1, T2, T3,... will be
1. 248√3
2. 192√3
3. 188√3
4. 164√3

Answer 1

Correct Answer - Option 2 : 192√3

Calculation:

Side of an equilateral triangle T1 = 1/2 × side of an equilateral triangle T2

Area of an equilateral triangle T1 = √3/4 × side²

Area of an equilateral triangle T2 = 1/4 × area of an equilateral triangle T1

Similarly,

Area of an equilateral triangle T3 = 1/16 × area of an equilateral triangle T1

Area of an equilateral triangle T1 = 24² × √3/4 = 144√3 sq.cm

Sum of such equilateral triangles = m + m/4 + m/16 + m/64 + ....

This is a geometric progression with first term = 144√3 sq.cm and common ratio r = 1/4

∴ Sum of a geometric progression = 144√3 / ( 1 - 1/4) = 192√3 sq.cm

Sum of infinite terms of geometric progression = a/(1 - r) where a = first term and r = common ratio