A line segment has endpoints at #(9 ,6 )# and #(5 ,3)#. If the line segment is rotated about the origin by #pi /2 #, translated vertically by #2 #, and reflected about the xaxis, what will the line segment's new endpoints be?
Since there are 3 transformations to be performed here, label the points A(9 ,6) and B(5 ,3) so that the change to each point after each transformation can be noted.
hence A(9 ,6) →A'(6 ,9) and B(5 ,3) → B'(3 ,5)
hence A'(6 ,9) → A''(6 ,7) and B'(3 ,5) → B''(3 ,3)
Third transformation Under a reflection in the xaxis
hence A''(6 ,7) → A'''(6 ,7) and B''(3 ,3) → B'''(3 ,3)
After all 3 transformations.
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To find the new endpoints of the line segment after the described transformations, we need to perform each transformation step by step:

Rotation by π/2 about the origin: To rotate a point (x, y) by an angle θ about the origin, the new coordinates (x', y') can be found using the formulas: [x' = x \cdot \cos(\theta)  y \cdot \sin(\theta)] [y' = x \cdot \sin(\theta) + y \cdot \cos(\theta)] Applying these formulas with θ = π/2, the original endpoints (9, 6) and (5, 3) become (6, 9) and (3, 5), respectively.

Translation vertically by 2: To translate a point (x, y) vertically by a distance d, the new coordinates become (x, y + d). Applying this translation, the new endpoints become (6, 11) and (3, 7), respectively.

Reflection about the xaxis: To reflect a point (x, y) about the xaxis, the ycoordinate becomes its negative, i.e., (x, y). After reflection, the new endpoints become (6, 11) and (3, 7), respectively.
So, after the described transformations, the line segment's new endpoints are (6, 11) and (3, 7).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 Points A and B are at #(3 ,8 )# and #(7 ,3 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #5 #. If point A is now at point B, what are the coordinates of point C?
 Points A and B are at #(9 ,3 )# and #(7 ,8 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #3 #. If point A is now at point B, what are the coordinates of point C?
 A triangle has corners at #(7 ,2 )#, #(3 ,1 )#, and #(5 ,7 )#. If the triangle is dilated by a factor of #2/5 # about point #(6 ,8 ), how far will its centroid move?
 Circle A has a radius of #4 # and a center of #(6 ,2 )#. Circle B has a radius of #3 # and a center of #(5 ,7 )#. If circle B is translated by #<2 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
 Circle A has a radius of #1 # and a center of #(1 ,7 )#. Circle B has a radius of #2 # and a center of #(8 ,1 )#. If circle B is translated by #<4 ,3 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
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