Points A and B are at #(4 ,1 )# and #(5 ,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} ): [ \begin{aligned} x' &= x \cos(\theta) - y \sin(\theta) \ y' &= x \sin(\theta) + y \cos(\theta) \end{aligned} ] Substituting the values for point A and ( \theta = \frac{3\pi}{2} ): [ \begin{aligned} x' &= 4 \cos\left(\frac{3\pi}{2}\right) - 1 \sin\left(\frac{3\pi}{2}\right) \ y' &= 4 \sin\left(\frac{3\pi}{2}\right) + 1 \cos\left(\frac{3\pi}{2}\right) \end{aligned} ] Evaluating the trigonometric functions: [ \begin{aligned} x' &= 4 \cdot 0 - 1 \cdot (-1) = 1 \ y' &= 4 \cdot (-1) + 1 \cdot 0 = -4 \end{aligned} ] So after rotation, point A is now at (1, -4).
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Dilation about Point C by a Factor of 4: The dilation formula for a point (x, y) about a center (h, k) by a factor of ( r ) is given by: [ \begin{aligned} x' &= h + r(x - h) \ y' &= k + r(y - k) \end{aligned} ] Substituting the values for point A (1, -4) and the dilation factor of 4, and knowing that point A becomes point B after dilation: [ \begin{aligned} 5 &= h + 4(1 - h) \ 9 &= k + 4(-4 - k) \end{aligned} ] Solving these equations gives the coordinates of point C: [ \begin{aligned} h &= 3 \ k &= 4 \end{aligned} ] Therefore, the coordinates of point C are (3, 4).
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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.
- A triangle has corners at #(-6 ,3 )#, #(3 ,-2 )#, and #(5 ,7 )#. If the triangle is dilated by a factor of #5 # about point #(-2 ,-1 ), how far will its centroid move?
- Point A is at #(8 ,-4 )# and point B is at #(2 ,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 triangle has corners at #(-2 ,3 )#, #(1 ,-4 )#, and #(5 ,-6 )#. If the triangle is dilated by a factor of #2 # about point #(-2 ,-3 ), how far will its centroid move?
- Points A and B are at #(9 ,4 )# and #(7 ,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 #(-1 ,2 )#, #(3 ,-5 )#, and #(7 ,4 )#. If the triangle is dilated by a factor of #5 # about point #(-2 ,6 ), how far will its centroid move?

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