# Points A and B are at #(2 ,2 )# and #(3 ,7 )#, respectively. Point A is rotated counterclockwise about the origin by #pi/2 # 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?

New coordinates of A after

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To find the coordinates of point C, we need to perform two transformations: rotation counterclockwise by (\frac{\pi}{2}) about the origin and dilation by a factor of 3 about point C.

First, let's perform the rotation counterclockwise by (\frac{\pi}{2}) about the origin. The rotation of a point (x, y) counterclockwise by (\frac{\pi}{2}) about the origin results in the point (-y, x).

So, applying this to point A(2, 2), we get (-2, 2).

Now, we need to dilate this point about point C by a factor of 3. Let's denote the coordinates of point C as (x, y).

The dilation formula is given by: [x' = x + k(x_c - x)] [y' = y + k(y_c - y)]

where (x, y) are the original coordinates, (x', y') are the new coordinates after dilation, (x_c, y_c) are the coordinates of the center of dilation, and k is the scale factor.

Given that the new coordinates after dilation are (3, 7), and the scale factor is 3, we can set up the equations:

[3 = -2 + 3(x_c + 2)] [7 = 2 + 3(y_c - 2)]

Solving these equations simultaneously will give us the coordinates of point C.

From the first equation: [3 = -2 + 3x_c + 6] [3 = 3x_c + 4] [3x_c = -1] [x_c = -\frac{1}{3}]

From the second equation: [7 = 2 + 3y_c - 6] [7 = 3y_c - 4] [3y_c = 11] [y_c = \frac{11}{3}]

So, the coordinates of point C are ((- \frac{1}{3}, \frac{11}{3})).

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The coordinates of point C are (2, 7).

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When evaluating a one-sided 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 one-sided 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 one-sided 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.

- A line segment has endpoints at #(5 , 8)# and #(6 , 1)#. If the line segment is rotated about the origin by #pi/2 #, translated vertically by #-2#, and reflected about the y-axis, what will the line segment's new endpoints be?
- Point A is at #(-2 ,-5 )# and point B is at #(-3 ,1 )#. Point A is rotated #pi/2 # 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?
- A triangle has corners at #(6 ,2 )#, #(5 ,-8 )#, and #(-5 ,3 )#. If the triangle is dilated by a factor of #5 # about point #(7 ,-2 ), how far will its centroid move?
- A triangle has corners at #(-2 ,1 )#, #(8 ,-5 )#, and #(-3 ,-2 )#. If the triangle is dilated by a factor of #5 # about point #(7 ,5 ), how far will its centroid move?
- Circle A has a radius of #3 # and a center of #(2 ,7 )#. Circle B has a radius of #4 # and a center of #(7 ,5 )#. If circle B is translated by #<-1 ,1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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