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?

2 Answers
Narad T.
Mar 21, 2017

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

Explanation:

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)#

Jim G.
Mar 21, 2017

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

Explanation:

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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