Question

Start with an equilateral triangle of side length 1 . Construct a second triangle by connecting the midpoints of the sides of the first triangle. Construct a third triangle by connecting the midpoints of the edges of the second triangle. Continue this process indefinitely. The sum of the areas of all the triangles, is

• A. $\sqrt{3}$
• B. $\frac{2}{\sqrt{3}}$
• C. $\frac{1}{\sqrt{3}}$
• D. $\infty$

Answer 1

Answer: (C)

Sum of area of all triangles $=\frac{\sqrt{3}}{4}\left(1^2+\left(\frac{1}{2}\right)^2+\left(\frac{1}{4}\right)^2+\ldots \ldots \infty\right)$
$=\frac{\sqrt{3}}{4}\left(\frac{1}{1-\frac{1}{4}}\right)=\frac{1}{\sqrt{3}}$