Points A and B are at #(9 ,7 )# and #(3 ,4 )#, respectively. Point A is rotated counterclockwise about the origin by #pi/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?
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To find the coordinates of point C, we'll follow these steps:
- Rotate point A counterclockwise about the origin by ( \frac{\pi}{2} ) to get the new coordinates of A.
- Since point A is now at point B after dilation about point C by a factor of 2, we'll find the midpoint between A and B, which represents point C.
Let's proceed:
- Rotate point A counterclockwise about the origin by ( \frac{\pi}{2} ). The rotation of a point ((x, y)) counterclockwise by an angle ( \theta ) about the origin is given by: [ (x', y') = (x \cos(\theta) - y \sin(\theta), x \sin(\theta) + y \cos(\theta)) ]
For point A(9, 7): [ (x', y') = (9 \cos(\frac{\pi}{2}) - 7 \sin(\frac{\pi}{2}), 9 \sin(\frac{\pi}{2}) + 7 \cos(\frac{\pi}{2})) ]
[ (x', y') = (7, -9) ]
- Now, we find the midpoint between the new point A(7, -9) and the original point B(3, 4) to get the coordinates of point C: [ C_x = \frac{7 + 3}{2} = 5 ] [ C_y = \frac{-9 + 4}{2} = -\frac{5}{2} ]
So, the coordinates of point C are (5, -\frac{5}{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.
- Point A is at #(1 ,6 )# and point B is at #(5 ,-3 )#. Point A is rotated #pi # 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 ,1 )#, #(8 ,-5 )#, and #(-1 ,-2 )#. If the triangle is dilated by a factor of #5 # about point #(4 ,-6 ), how far will its centroid move?
- A line segment with endpoints at #(5 , -8 )# and #(2, -2 )# is rotated clockwise by #(3 pi)/2#. What are the new endpoints of the line segment?
- In a photograph, a sculpture is 4.2 cm tall and 2.5 cm wide. The scale factor is #1/16#. What is the size of the actual sculpture?
- Point A is at #(-9 ,-4 )# and point B is at #(-5 ,-8 )#. 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?

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