Question

Points A and B are at `(2 ,6 )` and `(1 ,9 )`, respectively. Point A is rotated counterclockwise about the origin by `pi` and dilated about point C by a factor of `1/2`. If point A is now at point B, what are the coordinates of point C?

Answer 1

`C=(1,6)`

Explanation:

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

`• " a point "(x,y)to(-x,-y)`

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

`rArrvec(CB)=color(red)(1/2)vec(CA')`

`rArrulb-ulc=1/2(ula'-ulc)`

`rArrulb-ulc=1/2ula'-1/2ulc`

`rArr1/2ulc=ulb-1/2ula'`

`color(white)(rArr1/2ulc)=((1),(9))-1/2((-2),(-6))`

`color(white)(rArr1/2ulc)=((1),(9))-((-1),(-3))=((2),(12))`

`rArrulc=1/2((2),(12))=((1),(6))`

`rArrC=(1,6)`