Let A BC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle A BC and the same process is repeated infinitely many times. If P is the sum of

Question

Let A BC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle A BC and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is be the sum of areas of all the triangles formed in this process, then :

TestHub · Accepted Answer

Area of first Δ = 4 3 ​ a 2 ​ Area of second Δ = 4 3 ​ a 2 ​ 4 a 2 ​ = 16 3 ​ a 2 ​ Area of third Δ = 64 3 ​ a 2 ​ sum of area = 4 3 ​ a 2 ​ ( 1 + 4 1 ​ + 16 1 ​ … ) ​ Q = 4 3 ​ a 2 ​ 4 3 ​ 1 ​ = 3 ​ a 2 ​ perimeter of 1 st Δ = 3 a perimeter of 2 nd Δ = 2 3 a ​ perimeter of 3 rd Δ = 4 3 a ​ P = 3 a ( 1 + 2 1 ​ + 4 1 ​ + … ) P = 3 a ⋅ 2 = 6 a a = 6 P ​ Q = 3 ​ 1 ​ ⋅ 36 P 2 ​ P 2 = 36 3 ​ Q ​

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