Points A and B are at #(5 ,9 )# and #(8 ,6 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # 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 rotating point A counterclockwise by π radians about the origin and dilating it by a factor of 4 about point C to reach point B, follow these steps:
- Find the coordinates of the rotated point A.
- Use the coordinates of points A and B to find the coordinates of point C.
Let's proceed:
- To rotate point A counterclockwise by π radians about the origin, we can use the rotation matrix:
[ \begin{pmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{pmatrix} ]
For a counterclockwise rotation by π radians, the matrix becomes:
[ \begin{pmatrix} \cos(\pi) & -\sin(\pi) \ \sin(\pi) & \cos(\pi) \end{pmatrix} = \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} ]
Now, we multiply this rotation matrix by the coordinates of point A:
[ \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} \begin{pmatrix} 5 \ 9 \end{pmatrix} = \begin{pmatrix} -5 \ -9 \end{pmatrix} ]
- Now, we need to find the coordinates of point C, which acts as the center of dilation.
Since point A is dilated by a factor of 4 to reach point B, the coordinates of point C can be found using the midpoint formula:
[ C = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) ]
[ C = \left( \frac{5 + 8}{2}, \frac{9 + 6}{2} \right) = \left( \frac{13}{2}, \frac{15}{2} \right) ]
So, the coordinates of point C are ((\frac{13}{2}, \frac{15}{2})).
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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.
- CD has endpoints, C( -8, 3) and D (-8, -6) Rotate the segment about the origin, 90 clockwise. What are the coordinates of C' and D'?
- Circle A has a radius of #2 # and a center at #(3 ,1 )#. Circle B has a radius of #4 # and a center at #(8 ,3 )#. If circle B is translated by #<-4 ,-1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- Point A is at #(8 ,-1 )# and point B is at #(9 ,-7 )#. 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 ,5 )# and #(2 ,3 )#. If the line segment is rotated about the origin by #( 3 pi)/2 #, translated horizontally by # - 1 #, and reflected about the y-axis, what will the line segment's new endpoints be?
- A triangle as corners at #(5 ,2 )#, #(2 ,4 )#, and #(3 ,8 )#. If the triangle is dilated by a factor of #3 # about #(2 ,3 ), how far will its centroid move?

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