Points A and B are at #(3 ,7 )# and #(7 ,2 )#, 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?
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To find the coordinates of point C after rotating point A counterclockwise about the origin by ( \pi ) radians and dilating about point C by a factor of 5, follow these steps:
- Find the coordinates of the image of point A after rotating it counterclockwise about the origin by ( \pi ) radians.
- Use the dilation factor to find the coordinates of point C.
Step 1: Rotate point A counterclockwise about the origin by ( \pi ) radians: Let ( A(x, y) ) be the coordinates of point A. After rotating counterclockwise by ( \pi ) radians, the new coordinates ( A'(x', y') ) are given by: [ x' = x \cos(\pi) - y \sin(\pi) ] [ y' = x \sin(\pi) + y \cos(\pi) ]
Step 2: Use the dilation factor to find the coordinates of point C: Let ( C(c_x, c_y) ) be the coordinates of point C. Since point A is dilated about point C by a factor of 5, the coordinates of point C can be found using the following relation: [ x' = 5(x - c_x) + c_x ] [ y' = 5(y - c_y) + c_y ]
Now, substitute the coordinates of point A after rotation into these equations to solve for the coordinates of point C.
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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.
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.
- Points A and B are at #(3 ,9 )# and #(9 ,2 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #2 #. If point A is now at point B, what are the coordinates of point C?
- A triangle has corners at #(4, 6 )#, ( 1 , 7)#, and #( 3, -4)#. What will the new coordinates of the triangle be if it is reflected across the x-axis?
- A line segment has endpoints at #(2 , 2)# and #(5 , 4)#. If the line segment is rotated about the origin by #(pi)/2 #, translated vertically by #3#, and reflected about the y-axis, what will the line segment's new endpoints be?
- Circle A has a radius of #2 # and a center at #(8 ,3 )#. Circle B has a radius of #3 # and a center at #(3 ,2 )#. If circle B is translated by #<-2 ,6 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- Circle A has a radius of #2 # and a center at #(5 ,2 )#. Circle B has a radius of #5 # and a center at #(3 ,4 )#. 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?

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