Problem

Source: Puerto Rico TST 2022.2.4

Tags: geometry



Let's construct a family {Kn} of geometric figures following the pattern shown in pictures: where each hexagon (like the starting one) is constructed by cutting the two corners tops of a square, in such a way that the two figures removed are identical isosceles triangles, and the three resulting upper sides have the same length. Continuing like this, a pattern is produced with which we can build the figures Kn, for integer n0 . Then, we denote by Pn and An the perimeter and area of the figure Kn, respectively. If the side of square to build K0 measures x: (a) Calculate P0 and A0 (in terms of the length x). (b) Find an explicit formula for Pn, and for An, in terms of x and of n. Simplify your answers. (c) If P2022=A2022, find the measure of the six sides of the figure K0, in its simplest form.