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Points A and B are at #(9 ,9 )# and #(7 ,8 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #2 #. If point A is now at point B, what are the coordinates of point C?

1 Answer

Answer:

Point #C(-25, -26)#

Explanation:

From #A(9, 9)#
point A will be at #A'(x_a', y_a')=(-9, -9)# after rotation of #pi# whether by counterclockwise or clockwise.

The formula for dilation with center of dilation #C(x_c, y_c)# by a factor #k# is
#x_a''=k(x_a'-x_c)+x_c# and #y_a''=k(y_a'-y_c)+y_c#

where #(x_a'', y_a'')# is the coordinates of the final position of #A#

Given that #B# is at #(7, 8)#, therefore #x_a''=7# and #y_a''=8#

The coordinates of #C(x_c, y_c)# can now be computed with #k=2#
#x_a''=k(x_a'-x_c)+x_c#
#7=2(-9-x_c)+x_c#
#7=-18-2*x_c+x_c#
#7=-18-x_c#
#x_c=-25#

And
#y_a''=k(y_a'-y_c)+y_c#
#8=2(-9-y_c)+y_c#
#8=-18-2y_c+y_c#
#8=-18-y_c#
#y_c=-26#

#C(x_c, y_c)=(-25, -26)#

God bless....I hope the explanation is useful.