Points A and B are at #(4 ,1 )# and #(3 ,9 )#, 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, we first need to perform the rotation and dilation operations on point A.
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Rotation Counterclockwise by ( \frac{3\pi}{2} ): To rotate point A counterclockwise by ( \frac{3\pi}{2} ) about the origin, we use the following rotation matrix: [ \begin{pmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{pmatrix} ] Substituting ( \theta = \frac{3\pi}{2} ), we have: [ \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} ] Multiplying this matrix by the coordinates of point A, we get the coordinates of the rotated point.
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Dilation about Point C by a Factor of 4: After obtaining the coordinates of the rotated point, we dilate it by a factor of 4 about point C. This involves multiplying the coordinates of the rotated point by 4 and adding the coordinates of point C.
By performing these operations, we'll obtain 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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