Star-Burst Race
4 runners race straight out, at different angles, from the same starting location. After 10 seconds, their positions can be connected to make an irregular quadrilateral. After 20 seconds, their positions can again be connected to make another irregular quadrilateral. If all 4 runners run steadily (although each at a different speed), are the two quadrilaterals similar?
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1 solution
The four paths of the runners split the quadrilaterals into 4 triangles. By side-angle-side similarity, the two triangles defined by each pair of runners are similar. Therefore, the sides of the two quadrilaterals are in a common ratio, and the angles in the 4 corners are unchanged. Therefore, the two irregular quadrilaterals are definitely similar.