Triangular Pattern using sequences, series and trigonometry

The triangle [GHI] below is equilateral with side of length 1. The coordinates of G are (0,0) and GH is horizontal. The picture is constructed by drawing a new equilateral triangle with one vertex on each of the edges GH, GI and HI. The procedure is repeated recursively generating a sequence of triangles, so that the angle between one triangle and the one inside it is always the same. We have coloured the triangles alternating green and white. Point P indicates one vertex of the first new triangle. P is on the edge GI. Building on your previous work you will study the triangles in the region shaded and outlined in black in the diagram below. For the purposes of this exercise you will work with GP=0.2 1 – What are the lengths and angles of the largest green triangle in the shaded region? Note that this triangle has edge GP. Give the coordinates of point P. Calculate the area of this triangle. Give lengths to 2 d.p. and angles in degrees to 1 d.p.. 2 – Calculate the area of the shaded region either by adding up the areas of the white and green triangles inside it or by using the size of the white equilateral triangle and the centre (or an alternative approach). We suggest you use the fact that the sequences of consecutive triangles are scalings of each other to simplify your calculations. Use as much of your knowledge of sequences and series as possible.

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