Question

Points A and B are at `(4 ,6 )` and `(7 ,5 )`, respectively. Point A is rotated counterclockwise about the origin by `pi/2` and dilated about point C by a factor of `5`. If point A is now at point B, what are the coordinates of point C?

Answer 1

`(-37/4,15/4)`

Explanation:

Rotating Point A counterclockwise by `pi/2` will give you a point at A' `(-6,4)`

The distance between `A':(-6,4) and B:(7,5)` is `sqrt(13^2+1^2)=sqrt(170)`

The distance between A' and B must be five times the distance between A' and C

The vector to go from B to A' is `{-13,-1}`

1/4 of that vector is `{-13/4,-1/4}`

Apply that to A' to get `(-6-13/4,4-1/4)=(-37/4,15/4)`

Answer 2

`C=(-37/4,15/4)`

Explanation:

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

`• " a point "(x,y)to(-y,x)`

`rArrA(4,6)toA'(-6,4)" where A' is the image of A"`

`rArrvec(CB)=color(red)(5)vec(CA')`

`rArrulb-ulc=5(ula'-ulc)`

`rArrulb-ulc=5ula'-5ulc`

`rArr4ulc=5ula'-ulb`

`color(white)(rArr4ulc)=5((-6),(4))-((7),(5))`

`color(white)(rArrulc)=((-30),(20))-((7),(5))=((-37),(15))`

`rArrulc=1/4((-37),(15))=((-37/4),(15/4))`

`rArrC=(-37/4,15/4)`