Let A B C be an equilateral triangle.

Question

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 $$\mathrm{P}$$

Accepted Answer

$	riangleABC$ is an equilateral triangle having side $=a$ unit

Now, perimeter $=$ sum of all sides $=3a$

Area $=\frac{\sqrt{3}}{4}{a}^{2}$

Now, $\ln 	riangleDEF,DE=\frac{a}{2}=EF=DF$

$\begin{gathered} =3	imes \frac{a}{2}=\frac{3a}{2} \ =\frac{\sqrt{3}}{4}	imes {(\frac{a}{2})}^{2}=\frac{\sqrt{3}{a}^{2}}{16} \end{gathered}$

Now, $P=3a+\frac{3a}{2}+\frac{3a}{4}+\cdots$

$Q=\frac{\sqrt{3}}{4}{a}^{2}+\frac{\sqrt{3}}{16}{a}^{2}+\frac{\sqrt{3}}{64}{a}^{2}+\cdots$

$P=\frac{3a}{1-\frac{1}{2}}=3a	imes 2\Rightarrow P=6a\,$

$Q=\frac{\frac{\sqrt{3}}{4}{a}^{2}}{1-\frac{1}{4}}=\frac{4}{3}	imes \frac{\sqrt{3}}{4}{a}^{2}\, Q=\frac{\sqrt{3}}{3}{a}^{2}\,$

From equation (i) \& (ii)

$\begin{gathered} P=6a \ Q=\frac{\sqrt{3}}{3}{a}^{2} \ Q=\frac{\sqrt{3}}{3}	imes \frac{{P}^{2}}{36} \ {P}^{2}=36\sqrt{3}Q \end{gathered}$

Solutions · 1