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

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) √3 (B) (2/√3) (C) (1/√3) (D) ∞

Tardigrade Admin · Accepted Answer

Sum of area of all triangles =(√3/4)(12+((1/2))2+((1/4))2+ ldots ldots ∞) =(√3/4)((1/1-(1)4))=(1/√3)

$3$

$\frac{2}{3}$

$\frac{1}{3}$

$\infty$

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

3​

3​2​

3​1​

∞

Sum of area of all triangles =43​​(12+(21​)2+(41​)2+……∞) =43​​(1−41​1​)=3​1​

$3$

$\frac{2}{3}$

$\frac{1}{3}$

$\infty$

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