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 $\frac{π}{2}$ $pi/2$ 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?

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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 $\frac{π}{2}$ $pi/2$ 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?

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Answer 1 · 2016-04-07T15:46:00

$Point C \to P_{C} \to (x, y) \to (- 3 \frac{1}{2}, 8 \frac{1}{2})$ $color(green)("Point C "-> P_C ->(x,y)->(-3 1/2" "," "8 1/2))$

Explanation:

Tony B

$Point A_{1} is rotated through \frac{π}{2} to point A_{2}$ $"Point "A_1" is rotated through " pi/2" to point "A_2$

Points $C A_{2} and B$ $C" "A_2" and "B$ form a straight line

Distance C to B is 3 times the distance C to $A_{2}$ $A_2$

$Solved using ratios of triangle sides$ $color(red)("Solved using ratios of triangle sides")$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Determine x_{C}$ $color(blue)("Determine "x_C)$

$Taking us along the x-axis from B to A_{2}$ $color(brown)("Taking us along the x-axis from B to "A_2)$
$\Rightarrow x_{A_{2}} = x_{B} - (x_{B} - x_{A})$ $=> x_(A_2) = x_B -(x_B-x_A)$

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

But from B to C is $\frac{1}{2}$ $1/2$ as much again giving us the 3 halves.

$\Rightarrow x_{C} = x_{B} - (x_{B} - x_{A}) - (\frac{x_{B} - x_{A}}{2})$ $=> x_C =x_B -(x_B-x_A)-((x_B-x_A)/2)$

$\Rightarrow x_{C} = 1 - (3) - (1 \frac{1}{2}) = - 3 \frac{1}{3}$ $color(blue)(=>x_C= 1- (3)-(1 1/2) = -3 1/3)$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Determine y_{C}$ $color(blue)("Determine "y_C)$
$Taking us along the y-axis from B to A_{2}$ $color(brown)("Taking us along the y-axis from B to "A_2)$

$\Rightarrow y_{A_{2}} = y_{B} + (y_{A_{2}} - y_{B})$ $=>y_(A_2) = y_B +(y_(A_2)-y_B)$

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

But from B to C is $\frac{1}{2}$ $1/2$ as much again giving us the 3 halves.

$\Rightarrow y_{A_{2}} = y_{B} + (y_{A_{2}} - y_{B}) + \frac{y_{A_{2}} - y_{B}}{2}$ $=>y_(A_2) = y_B +(y_(A_2)-y_B)+(y_(A_2)-y_B)/2$

$\Rightarrow y_{A_{2}} = 7 + (1) + (\frac{1}{2}) = 8 \frac{1}{2}$ $color(blue)(=>y_(A_2) = 7+(1)+(1/2) = 8 1/2)$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$So point C \to P_{C} \to (x, y) \to (- 3 \frac{1}{2}, 8 \frac{1}{2})$ $color(green)("So point C "-> P_C ->(x,y)->(-3 1/2" "," "8 1/2))$

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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 $\frac{π}{2}$ $pi/2$ 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?

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Points A and B are at (8 ,2 )(8,2)(8 ,2 ) and (1 ,7 )(1,7)(1 ,7 ), respectively. Point A is rotated counterclockwise about the origin by pi/2 π2pi/2 and dilated about point C by a factor of 3 33 . If point A is now at point B, what are the coordinates of point C?

1 Answer

Explanation:

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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 $\frac{π}{2}$ $pi/2$ 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?