Question

Points A and B are at `(4 ,1 )` and `(7 ,5 )`, respectively. Point A is rotated counterclockwise about the origin by `pi/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=(15/4,7/2)`

Explanation:

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

`• " a point "(x,y)to(-y,x)`

`A(4,1)toA'(-1,4)" where A' is the image of A"`

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

`"expressing in terms of position vectors gives"`

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

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

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

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

`color(white)(1/2ulc)=((7),(5))-((-1/2),(-2))=((15/2),(7))`

`ulc=1/2((15/2),(7))=((15/4),(7/2))`

`"thus "C=(15/4,7/2)`