Question

One side of an equilateral triangle is $24$ cm. The midpoints of its sides are joined to form another triangle whose midpoints are in turn joined to form still another triangle this process continues indefinitely. The sum of the perimeters of all the triangles is

• A. $144\text{cm}$
• B. $142\text{cm}$
• C. $140\text{cm}$
• D. $130\text{cm}$

Answer 1

The correct option is A $144\text{cm}$


The mid-points of its sides are joined to from another triangle whose mid-points are joined to from another triangle. This process continues indefinitely.
So, side of $2$nd equilateral triangle $=12$ cm ( line segment joining mid-point of two sides of a triangle is parallel to third side and half of it ).
Similarly, side of third equilateral triangle $=6$ cm so on

The perimeter of all the triangles
$=3\left(24\right)+3\left(12\right)+3\left(6\right)+\dots \dots =\frac{3\left(24\right)}{1-\frac{1}{2}}=144\text{cm}$