In an equilateral triangle with a side of 8 cm, another triangle is inscribed with the vertices of which are the midpoints of the sides of the first; in the second triangle, a triangle is inscribed in the same way, and so on. Find the perimeter of the eighth triangle.
1) Consider first the first triangle after the main one. Each side of it is equal to half the side of the main triangle, and the perimeter is half the perimeter of the original triangle.
Let side a1 = 8 cm, perimeter p = 3 * a1 = 3 * 8 cm = 24 cm.
2) The next perimeter p2 = 3 * a2 = p1 / 2 = 24/2 = 12 cm. Next, we calculate the perimeter values, denoting the number of the triangle, we get:
3) p3 = p2 / 2 = 12 cm / 2 = 6 cm.
4) p4 = p3 / 2 = 6 cm / 2 = 3 cm.
5) p5 = p4 / 2 = 3/2 = 1.5 cm. 6) p8 = p1 / 2 ^ 8 = 24 cm / 256.
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