The midpoints of the three sides of an equilateral triangle are connected to form a second triangle. A third triangle is formed by connecting the midpoints of the second triangle. This process is repeated until a tenth triangle is formed. What is the ratio of the perimeter of the tenth triangle to that perimeter of the third triangle? By midpoints of the second triangle I mean the midpoints of it's sides.
+129849 The perimeter of each successive triangle is (1/2) of the previous triangle
Call the perimeter of the first triangle = P
The perimeter of the nth triangle is P (1/2)^(n - 1)
So...the perimeter of of the 10th triangle is P(1/2)*(10-1) = P(1/2)^9
The perimeter of the 3rd triangle is P(1/2)^(3 - 1) = P(1/2)^2
The ratio of the perimeter of the 10th triangle to the perimeter of the 3rd triangle is
(1/2)^9 / ( 1/2)^2 = (1/2)^7 = 1 / 128
