Question

Points A and B are at `(6 ,2 )` and `(3 ,8 )`, respectively. Point A is rotated counterclockwise about the origin by `pi/2` 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?

Answer 1

`color(indigo)("Coordinates of " C((x),(y)) = ((-11),(26))`

Explanation:

`A(9,7), B(3,8), "counterclockwise rotation "`pi/2`, "dilation factor" 2`

New coordinates of A after `(3pi)/2` counterclockwise rotation

`A(9,7) rarr A' (-7,9)`

`vec (BC) = (2) vec(A'C)`

`b - c = (2)a' - (2)c`

`c = (2)a' - b`

`C((x),(y)) = (2)((-7),(9)) + ((3),(8)) = ((-11),(26))`

Answer 2

`C=(-7,4)`

Explanation:

`"under a counterclockwise rotation about the origin of "pi/2`

`• " a point "(x,y)to(-y,x)`

`A(6,2)toA'(-2,6)" where A' is the image of A "`

`vec(CB)=color(red)(2)vec(CA')`

`ulb-ulc=2(ula'-ulc)`

`ulb-ulc=2ula'-2ulc`

`ulc=2ula'-ulb`

`color(white)(ulc)=2((-2),(6))-((3),(8))`

`color(white)(ulc)=((-4),(12))-((3),(8))=((-7),(4))`

`rArrC=(-7,4)`