Points A and B are at #(2 ,9 )# and #(7 ,5 )#, respectively. Point A is rotated counterclockwise about the origin by #(3pi)/2 # and dilated about point C by a factor of #1/2 #. 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 after the rotation and dilation, you can follow these steps:
- Rotate point A counterclockwise about the origin by ( \frac{3\pi}{2} ) radians. This rotation transforms point A into a new position.
- Dilate the rotated point A about point C by a factor of ( \frac{1}{2} ). This rescales the position of the point.
Let's first perform the rotation on point A:
Given point A: ( (2, 9) )
After rotating counterclockwise by ( \frac{3\pi}{2} ) radians, the new coordinates become:
( x' = x \cdot \cos(\frac{3\pi}{2}) - y \cdot \sin(\frac{3\pi}{2}) ) ( y' = x \cdot \sin(\frac{3\pi}{2}) + y \cdot \cos(\frac{3\pi}{2}) )
Substituting the coordinates of point A:
( x' = 2 \cdot \cos(\frac{3\pi}{2}) - 9 \cdot \sin(\frac{3\pi}{2}) = 9 ) ( y' = 2 \cdot \sin(\frac{3\pi}{2}) + 9 \cdot \cos(\frac{3\pi}{2}) = -2 )
Now, we have the coordinates after rotation: ( (9, -2) )
Next, let's dilate this point about an unknown point C by a factor of ( \frac{1}{2} ). Let the coordinates of point C be ( (x_C, y_C) ).
Using the dilation formula:
( x_C = \frac{x_A + x_B}{2} ) ( y_C = \frac{y_A + y_B}{2} )
Substituting the coordinates of point A (after rotation) and B:
( x_C = \frac{9 + 7}{2} = 8 ) ( y_C = \frac{-2 + 5}{2} = \frac{3}{2} )
Therefore, the coordinates of point C are ( (8, \frac{3}{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.
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