A side of an equilateral triangle is 24 cm long. A second equilateral triangle is inscribed in it by joining the midpoints of the sides of the first triangle; the process is continued. Find the perimeter of the sixth inscribed equilateral triangle.
A side of an equilateral triangle is 24 cm long. A second equilateral triangle is inscribed in it by joining the midpoints of the sides of the first triangle; the process is continued. Find the perimeter of the sixth inscribed equilateral triangle.

To Find: The perimeter of the sixth inscribed equilateral triangle.
Given: Side of an equilateral triangle is 24 cm long.

As 2nd triangle is formed by joining the midpoints of the sides of the first triangle whose
side is equal to 24cm
[As shown in the figure]
So Side of a 2nd equilateral triangle is 12 cm long [half of the first triangle side]
∴ Side of 2nd equilateral triangle = half of side of a 1st equilateral triangle
∴ Side of 3rd equilateral triangle = half of side of a 2nd equilateral triangle
∴ …………. and So on
Therefore, Side of 6th equilateral triangle = half of side of a 5th equilateral triangle

So, Perimeter of a 6th equilateral triangle is 3 times the side of a 6th equilateral triangle
[NOTE: Perimeter of the triangle is equal to the sum of all three sides of the triangle, and in case of an equilateral triangle all sides are equal]
So, Perimeter of 6th equilateral triangle = 3 × 0.75 = 2.25 cm