Question

Given an equilateral triangle $T_1$ with side 24 cm, a second triangle $T_2$ is formed by joining the midpoints of the sides of $T_1$. Then a third triangle $T_3$ is formed by joining the midpoints of the sides of $T_2$. If this process of forming triangles is continued, the sum of the areas, in sq cm, of infinitely many such triangles $T_1, T_2, T_3, \dots$ will be

Accepted Answer

A. Area of $T_1 = \frac{\sqrt{3}}{4} 	imes 24^2 = 144\sqrt{3}$. Joining midpoints reduces area by a factor of $\frac{1}{4}$. So areas form an infinite GP with $a = 144\sqrt{3}$ and $r = \frac{1}{4}$. Sum = $\frac{a}{1 - r} = \frac{144\sqrt{3}}{1 - 1/4} = \frac{144\sqrt{3}}{3/4} = 192\sqrt{3}$.

Infinite Equilateral Triangles Area Sum

Explanation

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