Question

Points A and B are at `(4 ,5 )` and `(6 ,2 )`, respectively. Point A is rotated counterclockwise about the origin by `(3pi)/2` 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=(7/4,2)`

Explanation:

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

`• " a point "(x,y)to(y,-x)`

`rArrA(4,5)toA'(5,-4)" 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)=((6),(2))-1/2((5),(-4))`

`color(white)(rArr1/2ulc)=((6),(2))-((5/2),(-2))=((7/2),(4))`

`rArrulc=1/2((7/2),(4))=((7/4),(2))`

`rArrC=(7/4,2)`