Points A and B are at #(1 ,5 )# and #(2 ,3 )#, 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?

Answer 1

#C=(6,1/3)#

#"under a counterclockwise rotation about the origin of "(3pi)/2#
#• " a point "(x,y)to(y,x)#
#rArrA(1,5)toA'(5,1)" where A' is the image of A"#
#rArrvec(CB)=color(red)(4)vec(CA')#
#rArrulb-ulc=4(ula'-ulc)#
#rArrulb-ulc=4ula'-4ulc#
#rArr3ulc=4ula'-ulb#
#color(white)(rArr3ulc)=4((5),(1))-((2),(3))#
#color(white)(rArr3ulc)=((20),(4))-((2),(3))=((18),(1))#
#rArrulc=1/3((18),(1))=((6),(1/3))#
#rArrC=(6,1/3)#
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Answer 2

To find the coordinates of point C after point A is rotated counterclockwise about the origin by ( \frac{3\pi}{2} ) and dilated about point C by a factor of 4 to reach point B, follow these steps:

  1. Find the coordinates of point A after rotating it counterclockwise about the origin by ( \frac{3\pi}{2} ).
  2. Use the dilation factor of 4 to determine the new coordinates of point A after dilation.
  3. Set up equations using the coordinates of points A and B to find the coordinates of point C.

Let's proceed:

  1. Rotating point A counterclockwise by ( \frac{3\pi}{2} ) swaps the coordinates ( (x, y) ) to ( (-y, x) ). So, after rotation, the coordinates of point A become ( (-5, 1) ).
  2. The dilation factor of 4 means that each coordinate of point A will be multiplied by 4. So, after dilation, the coordinates of point A become ( (-20, 4) ).
  3. Now, we set up equations using the coordinates of points A and B: ( -20 + h = 2 ) (for x-coordinates) ( 4 + k = 3 ) (for y-coordinates) Solve these equations to find ( h ) and ( k ), which represent the x-coordinate and y-coordinate of point C, respectively.

Solving these equations, we find: ( h = 22 ) ( k = -1 )

So, the coordinates of point C are ( (22, -1) ).

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Answer from HIX Tutor

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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