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Turning Triangles

A triangle ABC resting on a horizontal line is "rolled" along the line. Describe the paths of each of the vertices and the relationships between them and the original triangle.

Rolling Triangle

Age 11 to 14 Challenge Level:

Brian sent us his thoughts on this problem:

What happens: It is impossible to tell from the motion of the plank that the roller is not circular. The plank moves forwards at a steady rate (assuming a steady push!) and remains at a constant height (the side length of the triangle) above the floor.

Rolling triangle

Why: Consider a starting point with C on the floor and the plank parallel to the floor and touching the roller at B (BC is vertical). Move the plank, rolling the roller, until the plank is touching the roller at A (so now AC is vertical). The roller will be rotating about C. Clearly, given that curve AB is equidistant from C, the plank will remain at a constant height above the floor. The same arguments can be used if a vertex is against the plank and the circular section is against the floor; only the frame of reference is changed and relative motion of plank and floor are the same.At the point of transition, both cases are simultaneously true.
Now consider the transition. In this example, C will be on the floor, A will be against the plank, and any further movement will result in A being effectively fixed on the plank and curve BC rolling on the floor. Note that the tangent to curve BC at C is parallel to the tangent to curve AB at A --- AC is a common radius of both curves. Hence at the transition there will be smooth transfer of movement from one mode to the other. Despite one's intuitive notions, the roller will therefore have a smooth forward motion!