# Point A is at #(9 ,-2 )# and point B is at #(2 ,4 )#. Point A is rotated #pi # clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?

The distance has changed by

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The new coordinates of point A after rotating π radians clockwise about the origin are (-9, 2). The distance between points A and B remains unchanged after the rotation. Therefore, the distance between the original point A and point B is equal to the distance between the rotated point A and point B. The distance between points A and B can be calculated using the distance formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}). Substituting the coordinates of the points into the formula, the distance between points A and B is found to be (d = \sqrt{(2 - 9)^2 + (4 - (-2))^2} = \sqrt{(-7)^2 + (6)^2} = \sqrt{49 + 36} = \sqrt{85}). Therefore, the distance between points A and B remains (\sqrt{85}) units.

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- Circle A has a radius of #3 # and a center of #(3 ,9 )#. Circle B has a radius of #2 # and a center of #(1 ,4 )#. If circle B is translated by #<3 ,-1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
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