# Points A and B are at #(1 ,5 )# and #(2 ,3 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #4 #. If point A is now at point B, what are the coordinates of point C?

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To find the coordinates of point C after point A is rotated counterclockwise about the origin by ( \frac{3\pi}{2} ) and dilated about point C by a factor of 4 to reach point B, follow these steps:

- Find the coordinates of point A after rotating it counterclockwise about the origin by ( \frac{3\pi}{2} ).
- Use the dilation factor of 4 to determine the new coordinates of point A after dilation.
- Set up equations using the coordinates of points A and B to find the coordinates of point C.

Let's proceed:

- Rotating point A counterclockwise by ( \frac{3\pi}{2} ) swaps the coordinates ( (x, y) ) to ( (-y, x) ). So, after rotation, the coordinates of point A become ( (-5, 1) ).
- The dilation factor of 4 means that each coordinate of point A will be multiplied by 4. So, after dilation, the coordinates of point A become ( (-20, 4) ).
- Now, we set up equations using the coordinates of points A and B: ( -20 + h = 2 ) (for x-coordinates) ( 4 + k = 3 ) (for y-coordinates) Solve these equations to find ( h ) and ( k ), which represent the x-coordinate and y-coordinate of point C, respectively.

Solving these equations, we find: ( h = 22 ) ( k = -1 )

So, the coordinates of point C are ( (22, -1) ).

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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 triangle has corners at #(1, 3 )#, ( 8, -4)#, and #(1, -5 )#. If the triangle is reflected across the x-axis, what will its new centroid be?
- Circle A has a radius of #5 # and a center of #(8 ,2 )#. Circle B has a radius of #3 # and a center of #(3 ,7 )#. If circle B is translated by #<2 ,-1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- Point A is at #(-2 ,-4 )# and point B is at #(-3 ,6 )#. Point A is rotated #(3pi)/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 line segment has endpoints at #(4 ,0 )# and #(2 ,1 )#. If the line segment is rotated about the origin by #( pi)/2 #, translated vertically by #-8 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- Point A is at #(9 ,-2 )# and point B is at #(1 ,-3 )#. 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?

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