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Question 9 :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. 192√3
  2. 164√3
  3. 248√3
  4. 188√3

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Explanatory Answer

Method of solving this CAT Question from Geometry
Geometry - Infinite Triangles

As the triangle progresses infinitely and the side length decreases, it follows an infinite GP series
As the sides decrease by half, their areas decrease by \\frac{1}{4})
We know, Area of an Equilateral Triangle = \\frac{√3}{4}) × a2
Area of T1 = \\frac{√3}{4}) × 24 × 24 = 144 √3 sq cms
Sum of an Infinite GP = \\frac{𝑎}{1−𝑟}) where a = 144 √3 , r = \\frac{1}{4})
Sum of areas ( T1, T2, T3,..) = \\frac{144√3}{1 - \frac{1}{4}}) = \\frac{4 × 144 √3}{3})
Therefore, Sum of areas= 192 √3 sq cms.

The question is " 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, Find the sum of the areas, in sq cm, of infinitely many such triangles T1, T2, T3,.. Then the sum of areas will be equal to 192 √3 sq cms "

Hence, the answer is 192 √3 sq cms.

Choice A is the correct answer.

 

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