you start with an equilateral triangle and then trisect each of the sides with a new equilateral triangle on the middle third of each trisection, repeat the process indefinitely
how would you prove that the perimeter is infinite?
You start out with three sides of length a, and the figure perimeter is 3a.
After the first step you described, you have a figure with 12 sides of length a/3. The perimeter is now 4a.
After a similar second step, you will have 48 sides of length a/9. The perimeter is now 5.33 a
After N such steps, you have 3*4^N sides of length a/3^N, with a perimeter of = 3a*(4/3)^N
That number clearly becomes infinite as N becomes infinite
To prove that the perimeter of the figure formed by the repeated process described is infinite, we can use a geometric series.
Let's consider the length of the sides of the equilateral triangles after each iteration. In the initial equilateral triangle, let's assume that the side length is 's'.
After the first iteration, each of the three trisection equilateral triangles will have side length 's/3'. The total length of these smaller triangles is 3*(s/3) = s.
After the second iteration, each of the nine trisection equilateral triangles will have side length (s/3)/3 = s/9. The total length of these smaller triangles is 9*(s/9) = s.
Following this pattern, after every iteration, the total perimeter of the triangles formed will always be s. Thus, no matter how many iterations we continue, the total perimeter will always be s, which is a constant value.
Therefore, it is evident that the perimeter of the figure formed by the repeated process described is not infinite, but rather a constant value.