Question

Points A and B are at `(2 ,2 )` and `(3 ,7 )`, respectively. Point A is rotated counterclockwise about the origin by `pi/2` and dilated about point C by a factor of `3`. If point A is now at point B, what are the coordinates of point C?

Answer 1

`color(maroon)("Coordinates of point C " (-9/2, -1/2)`

Explanation:

`A(2,2), B(3,7), "counterclockwise rotation "`pi/2`, "dilation factor" 3`

New coordinates of A after `(3pi)/2` counterclockwise rotation

`A(2,2) rarr A' (-2,2)`

`vec (BC) = (3) vec(A'C)`

`b - c = (3)a' - (3)c`

`2c = (3)a' - b`

`c = (3/2)a' -(1/2) b`

`C((x),(y)) = (3/2)((-2),(2)) - (1/2) ((3),(7)) = ((-9/2),(-1/2))`

Answer 2

`C=(-9/2,-1/2)`

Explanation:

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

`• " a point "(x,y)to(-y,x)`

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

`vec(CB)=color(red)(3)vec(CA')`

`ulb-ulc=3(ula'-ulc)`

`ulb-ulc=3ula'-3ulc`

`2ulc=3ula'-ulb`

`color(white)(2ulc)=3((-2),(2))-((3),(7))`

`color(white)(2ulc)=((-6),(6))-((3),(7))=((-9),(-1))`

`ulc=1/2((-9),(-1))=((-9/2),(-1/2))`

`rArrC=(-9/2,-1/2)`