Question

Points A and B are at `(4 ,7 )` and `(3 ,9 )`, respectively. Point A is rotated counterclockwise about the origin by `(3pi)/2` and dilated about point C by a factor of `4`. If point A is now at point B, what are the coordinates of point C?

Answer 1

`color(indigo)("Coordinates of "C((x),(y)) = ((19),(-43))`

Explanation:

`A(4,7), B(3,9), "counterclockwise rotation "`(3pi)/2`, "dilation factor" 4`

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

`A(4,7) rarr A' (7,-4)`

`vec (BC) = (4) vec(A'C)`

`b - c = (4)a' - (4)c`

`c = (4)a' - (3)b`

`color(indigo)(C((x),(y)) = (4)((7),(-4)) - (3)((3),(9)) = ((19),(-43))`

Answer 2

`C=(25/3,-25/3)`

Explanation:

`"under a counterclockwise rotation about the origin of "(3pi)/2`

`• " a point "(x,y)to(y,-x)`

`A(4,7)toA'(7,-4)" where A' is the image of A"`

`vec(CB)=color(red)(4)vec(CA')`

`ulb-ulc=4(ula'-ulc)`

`ulb-ulc=4ula'-4ulc`

`3ulc=4ula'-ulb`

`color(white)(3ulc)=4((7),(-4))-((3),(9))`

`color(white)(3ulc)=((28),(-16))-((3),(9))=((25),(-25))`

`ulc=1/3((25),(-25))=((25/3),(-25/3))`

`rArrC=(25/3,-25/3)`