Question

Points A and B are at `(2 ,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

`color(red)("Coordinates of " C (7,6)`

Explanation:

`A(2,5), B(6,2), "rotation "`(3pi)/2`, "dilation factor" 1/2`

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

`A(2,5) rarr A' (5,-2)`

`vec (BC) = (1/2) vec(A'C)`

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

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

`(1/2)C((x),(y)) = -(1/2)((5),(-2)) + ((6),(2)) = ((7/2),(3))`

`color(red)("Coordinates of " 2 *C ((7/2),3) = C(7,6)`

Answer 2

`C=(7,6)`

Explanation:

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

`• " a point "(x,y)to(y,-x)`

`A(2,5)toA'(5,-2)" where A' is the image of A"`

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

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

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

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

`color(white)(1/2ulc)=((6),(2))-1/2((5),(-2))`

`color(white)(1/2ulc)=((6),(2))-((5/2),(-1))=((7/2),(3))`

`ulc=2((7/2),(3))=((7),(6))`

`rArrC=(7,6)`