Let $A B C$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $A B C$ and the same process is repeated infinitely many times. If $\mathrm{P}$ is the sum of perimeters and $Q$ is be the sum of areas of all the triangles formed in this…

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

Aicharya · Accepted Answer

Area of first $\Delta=\frac{\sqrt{3} \mathrm{a}^2}{4}$ Area of second $\Delta=\frac{\sqrt{3} \mathrm{a}^2}{4} \frac{\mathrm{a}^2}{4}=\frac{\sqrt{3} \mathrm{a}^2}{16}$ Area of third $\Delta=\frac{\sqrt{3} \mathrm{a}^2}{64}$ sum of area $=\frac{\sqrt{3} \mathrm{a}^2}{4}\left(1+\frac{1}{4}+\frac{1}{16} \ldots\right)$ $\begin{aligned} & \mathrm{Q}=\frac{\sqrt{3} \mathrm{a}^2}{4} \frac{1}{\frac{3}{4}}=\frac{\mathrm{a}^2}{\sqrt{3}} \\ & \text { perimeter of } 1^{\text {st }} \Delta=3 \mathrm{a} \\ & \text { perimeter of } 2^{\text {nd }} \Delta=\frac{3 \mathrm{a}}{2} \\ & \text { perimeter of } 3^{\text {rd }} \Delta=\frac{3 \mathrm{a}}{4} \\ & \mathrm{P}=3 \mathrm{a}\left(1+\frac{1}{2}+\frac{1}{4}+\ldots\right) \\ & \mathrm{P}=3 \mathrm{a} \cdot 2=6 \mathrm{a} \\ & \mathrm{a}=\frac{\mathrm{P}}{6} \\ & \mathrm{Q}=\frac{1}{\sqrt{3}} \cdot \frac{\mathrm{P}^2}{36} \\ & \mathrm{P}^2=36 \sqrt{3} \mathrm{Q}\end{aligned}$

Let $A B C$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides…

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