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?

Answer 1

The point #C=(-28/3, -14)#

The matrix of a rotation counterclockwise by #pi# about the origin is
#((-1,0),(0,-1))#
Therefore, the transformation of point #A# is
#A'=((-1,0),(0,-1))((5),(9))=((-5),(-9))#
Let point #C# be #(x,y)#, then
#vec(CB)=4vec(CA')#
#((8-x),(6-y))=4((-5-x),(-9-y))#

So,

#8-x=4(-5-x)#
#8-x=-20-4x#
#3x=-28#
#x=-28/3#

and

#6-y=4(-9-y)#
#6-y=-36-4y#
#3y=-42#
#y=-14#

Therefore,

The point #C=(-28/3,-14)#
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Answer 2

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:

  1. Find the coordinates of the rotated point A.
  2. Use the coordinates of points A and B to find the coordinates of point C.

Let's proceed:

  1. 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} ]

  1. 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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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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