Question

Points A and B are at `(2 ,9 )` and `(7 ,5 )`, 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=(23,8)`

Explanation:

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

`• " a point "(x,y)to(-y,x)`

`rArrA(2,9)toA'(-9,2)" 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)=((7),(5))-1/2((-9),(2))`

`color(white)(rArr1/2ulc)=((7),(5))-((-9/2),(1))=((23/2),(4))`

`rArrulc=2((23/2),(4))=((23),(8))`

`rArrC=(23,8)`