Triangles: the base case
The first example, familiar to most students, concerns a triangle. The midpoint polygon divides the triangular region up into four congruent triangles so the area ratio is 1/4. Each edge of the midpoint triangle is half the length of its corresponding edge on the original, so the perimeter ratio is 1/2.
Demonstration 1. Area ratio for midpoint triangles
(Click on the picture or caption to go to the demonstration.)