A side of an equilateral triangle is 20cm long. A second equilateral triangle is inscribed in it by joining the mid points of the sides of the first triangle. The process is continued as shown in the accompanying diagram. Find the perimeter of the sixth inscribed equilateral triangle.

Solution:

Say ABC is a triangle with cm

Let’s say that D, E and F are respectively the midpoints of AC, CB and AB  that are joined to form an equilateral triangle named as DEF

We now need to find the length of side of ΔDEF

Now let us consider ΔCDE

cm … D is the midpoint of AC and E is the midpoint of CB

ΔCDE is isosceles

… isosceles triangle’s base angle

But  …∠ABC is equilateral

Therefore, ΔCDE is equilateral

As a result, cm

In the similar way, it can be shown that cm

Therefore, the series so formed of the equilateral triangle’s sides will be 20, 10, 5 …

The series so formed is Geometric Progression with the first term and common ratio

To find the perimeter of 6^th triangle inscribed we first have to find the side of 6^th triangle that is the 6^th term in the series

n^th term in GP is given by

As a result, the side of inscribed 6^th equilateral triangle is 5/8 cm and therefore its perimeter would be thrice the length of its side since it’s an equilateral triangle

The perimeter of the inscribed 6^th equilateral triangle is cm