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.

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

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

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

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