Points A and B are at #(8 ,2 )# and #(1 ,7 )#, respectively. Point A is rotated counterclockwise about the origin by #pi/2 # and dilated about point C by a factor of #3 #. If point A is now at point B, what are the coordinates of point C?

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Answer 1 · 2016-04-07T15:46:00

#color(green)("Point C "-> P_C ->(x,y)->(-3 1/2" "," "8 1/2))#

Tony B

#"Point "A_1" is rotated through " pi/2" to point "A_2#

Points #C" "A_2" and "B# form a straight line

Distance C to B is 3 times the distance C to #A_2#

#color(red)("Solved using ratios of triangle sides")#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine "x_C)#

#color(brown)("Taking us along the x-axis from B to "A_2)#
#=> x_(A_2) = x_B -(x_B-x_A)#

#color(brown)("Taking us along the x-axis from B to C")#

But from B to C is #1/2# as much again giving us the 3 halves.

#=> x_C =x_B -(x_B-x_A)-((x_B-x_A)/2)#

#color(blue)(=>x_C= 1- (3)-(1 1/2) = -3 1/3)#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine "y_C)#
#color(brown)("Taking us along the y-axis from B to "A_2)#

#=>y_(A_2) = y_B +(y_(A_2)-y_B)#

#color(brown)("Taking us along the y-axis from B to C")#

But from B to C is #1/2# as much again giving us the 3 halves.

#=>y_(A_2) = y_B +(y_(A_2)-y_B)+(y_(A_2)-y_B)/2#

#color(blue)(=>y_(A_2) = 7+(1)+(1/2) = 8 1/2)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(green)("So point C "-> P_C ->(x,y)->(-3 1/2" "," "8 1/2))#