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In the diagram below, Point A is \sqrt 3 units above O.  Whenever Tasha moves a point, she translates the point $\sqrt 2$ units to the right.  Whenever Richard moves a point, he rotates the point clockwise about point $O$ by $60^\circ.$

 

In one round of transformations, Tasha moves $A$ to $A',$ and then Richard moves $A'$ to $A''.$  In another round of transformations, Richard moves $A$ to $A_1,$ and then Tasha moves $A_1$ to $A_2.$  Find $A''A_2.$

 Dec 31, 2024
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The distance A''A_2 is equal to sqrt(7) - 1.

 Dec 31, 2024

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