Problematic triangle
Level
pending
Let G the intersection point of the medians of a triangle ABC with area 3 u.a. Consider the points A', B' and C' obtained by a 180º rotation from the A, B and C points, respectively, around G. Found the formed area by the union of the regions bounded by triangle ABC and A'B'C'.
The answer is 4.
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1 solution
The 180° rotation will keep all sides parallel to themselves. i.e. AB || A’B’, BC || B’C’, AC || A’C’ and the medians will align co-linearly because they pass through the center of rotation.
Moreover, since it is rotated about the centroid (point where the medians meet), side B’C’ will intersect sides AC and AB in 1:2 and so on. So any triangular 'spike' of the star has only 1/3rd of the original triangle’s height and thereby 1/9th the original area.
This happens for all the six triangles. So 1/3rd area of the original triangles is in the spikes, while 2/3rd area is overlapped. So the area of the star formed = 3 + 3 − 3 2 × 3 = 4