Question

Points A and B are at `(2 ,1 )` and `(4 ,7 )`, respectively. Point A is rotated counterclockwise about the origin by `(3pi)/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?

Answer 1

The coordinates of point `C=(-1/2,-13/2)`

Explanation:

The matrix of a rotation counterclockwise by `3/2pi` about the origin is

`((0,1),(-1,0))`

Therefore, the transformation of point `A` is

`A'=((0,1),(-1,0))((2),(1))=((1),(-2))`

Let point `C` be `(x,y)`, then

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

`((4-x),(7-y))=3((1-x),(-2-y))`

So,

`4-x=3(1-x)`

`4-x=3-3x`

`2x=-1`

`x=-1/2`

and

`7-y=3(-2-y)`

`7-y=-6-3y`

`2y=-6-7`

`y=-13/2`

Therefore,

point `C=(-1/2,-13/2)`

Answer 2

`C=(-1/2,-13/2)`

Explanation:

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

`• " a point " (x,y)to(y,-x)`

`rArrA(2,1)toA'(1,-2)" where A' is the image of A"`

`"under a dilatation about C of factor 3"`

`vec(CB)=color(red)(3)vec(CA')`

`rArrulb-ulc=color(red)(3)(ula'-ulc)`

`rArrulb-ulc=3ula'-3ulc`

`rArr2ulc=3ula'-ulb`

`color(white)(rArr2ulc)=3((1),(-2))-((4),(7))`

`color(white)(rArr2ulc)=((3),(-6))-((4),(7))=((-1),(-13))`

`rArrulc=1/2((-1),(-13))=((-1/2),(-13/2))`

`"the components of " ulc" are the coordinates of C"`

`rArrC=(-1/2,-13/2)`