Question

A side of an equilateral triangle is 20 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 as shown in the accompanying diagram. Find the perimeter of the sixth inscribed equilateral triangle.

Answer 1

The side of the first equilateral ΔABC = 20 cm

By joining the midpoints of the sides of this triangle

We get the second equilateral triangle which each side

= `20/2`

= 10 cm

[∵ The line joining the mid-points of two sides of a triangle is `1/2` and parallel to third side of the triangle]

Similarly each side of the third equilateral triangle = `10/2` = 5cm

∴ Perimeter of first triangle = 20 × 3 = 60 cm

Perimeter of the second triangle = 10 × 3 = 30 cm

And the perimeter of the third triangle = 5 × 3 = 15 cm

Therefore, the series will be 60, 30, 15, …

Which is G.P. in which a = 60

And r = `30/60 = 1/2`

Now, we have to find the perimeter of the sixth inscribed equilateral triangle

∴ a₆ = ar^6–1

= `60 xx (1/2)^5`

= `60 xx 1/32`

= `15/8` cm

Hence, the required perimeter = `15/8` cm.