Points A and B are at #(3 ,7 )# and #(7 ,2 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #5 #. If point A is now at point B, what are the coordinates of point C?

Answer 1

#C=(-11/2,-37/4)#

Under a counterclockwise rotation about the origin of #pi#
#• "a point " (x,y)to(-x,-y)#
#rArrA(3,7)toA'(-3,-7)# where A' is the image of A.
#" Under a dilatation about C of factor 5"#
Taking a #color(blue)"vector approach"#
#rArrvec(CB)=5vec(CA')#
#rArrulb-ulc=5(ula'-ulc)#
#rArrulb-ulc=5ula'-5ulc#
#rArr4ulc=5ula'-ulb#
#color(white)(rArr4c)=5((-3),(-7))-((7),(2))#
#color(white)(rArr4c)=((-15),(-35))-((7),(2))#
#color(white)(rArr4c)=((-22),(-37))#
#rArrulc=1/4((-22),(-37))=((-11/2),(-37/4))#
#rArrC=(-11/2,-37/4)#
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Answer 2

The point #C# is #(-1,-24/5)#

The point #A'# is symmetric about the origin #O#
The coordinates of #A'# is #=(-3,-7)#
Let the point #C# be #(x,y)#

Then,

#vec(A'B)=5*vec(A'C)#
#vec(A'B) =<7-(-3),2-(-7)>=<10,9>#
#vec(A'C) = < x-(-3),y-(-7)> = < x+3,y+7>#

Therefore,

#5*< x+3,y+7> = <10,9>#

So,

#5(x+3)=10#
#5x+15=10#
#5x=-5#
#x=-1#

and

#5(y+7)=9#
#5y+35=9#
#5y=9-35=-24#
#y=-24/5#

Therefore,

The point #C# is #(-1,-24/5)#
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Answer 3

To find the coordinates of point C after rotating point A counterclockwise about the origin by ( \pi ) radians and dilating about point C by a factor of 5, follow these steps:

  1. Find the coordinates of the image of point A after rotating it counterclockwise about the origin by ( \pi ) radians.
  2. Use the dilation factor to find the coordinates of point C.

Step 1: Rotate point A counterclockwise about the origin by ( \pi ) radians: Let ( A(x, y) ) be the coordinates of point A. After rotating counterclockwise by ( \pi ) radians, the new coordinates ( A'(x', y') ) are given by: [ x' = x \cos(\pi) - y \sin(\pi) ] [ y' = x \sin(\pi) + y \cos(\pi) ]

Step 2: Use the dilation factor to find the coordinates of point C: Let ( C(c_x, c_y) ) be the coordinates of point C. Since point A is dilated about point C by a factor of 5, the coordinates of point C can be found using the following relation: [ x' = 5(x - c_x) + c_x ] [ y' = 5(y - c_y) + c_y ]

Now, substitute the coordinates of point A after rotation into these equations to solve for the coordinates of point C.

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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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