Question

One side of an equilateral triangle is 24 cm. 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. Find the sum of the perimeters of all the triangles.

Answer 1

Perimeter of the largest (outermost) equilateral triangle = 3 × 24 = 72 cm. 

Now, the perimeter of the triangle formed by joining the midpoints of a given triangle will be half the perimeter of the original triangle. 

∴ Required sum = 72 + 36 + 18 + ............. upto infinite terms 

This is an infinite GP, where first term a = 72 and common ratio r = \(\frac{1}{2}.\)

∴ Required sum = \(\frac{a}{1-r}\) = \(\frac{72}{1-\frac{1}{2}}\) = \(\frac{72}{\frac{1}{2}}\) = 72 x 2 = 144 cm.