Question

One side of an equilateral triangle is $24cm$. The mid-points of its sides are joined to form another triangle whose mid-points are in turn joined to form still another triangle. This process continues indefinitely. Then the sum of the perimeters of all the triangles is?

• A. $144cm$
• B. $213cm$
• C. $288cm$
• D. $172cm$

Answer 1

Answer: (A)

$144cm$

When a triangle is formed by joining the midpoints of a triangle, its side-lengths will be half of the original triangle. Therefore, its perimeter will also be halved.
Perimeter of original triangle: $24x3=72$
Perimeter of next triangle: $\frac{72}{2}=36$
Perimeter of next triangle: $\frac{36}{2}=18$ and so on...
Thefore, total perimeter: $72+36+18+...=72(1+\frac{1}{2}+\frac{1}{4}+...)=72(\frac{1}{1-\frac{1}{2}})=144$
Formula used:
$a+ar+a{r}^2+a{r}^3+...=\frac{a}{1-r}$ for $|r|<1$