Question

Points A and B are at `(3 ,7 )` and `(7 ,2 )`, respectively. Point A is rotated counterclockwise about the origin by `pi` 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

The point `C` is `(-1,-24/5)`

Explanation:

The point `A'` is symmetric about the origin `O`

The coordinates of `A'` is `=(-3,-7)`

Let the point `C` be `(x,y)`

Then,

`vec(A'B)=5*vec(A'C)`

`vec(A'B) =<7-(-3),2-(-7)>=<10,9>`

`vec(A'C) = < x-(-3),y-(-7)> = < x+3,y+7>`

Therefore,

`5*< x+3,y+7> = <10,9>`

So,

`5(x+3)=10`

`5x+15=10`

`5x=-5`

`x=-1`

and

`5(y+7)=9`

`5y+35=9`

`5y=9-35=-24`

`y=-24/5`

Therefore,

The point `C` is `(-1,-24/5)`

Answer 2

`C=(-11/2,-37/4)`

Explanation:

Under a counterclockwise rotation about the origin of `pi`

`• "a point " (x,y)to(-x,-y)`

`rArrA(3,7)toA'(-3,-7)` where A' is the image of A.

`" Under a dilatation about C of factor 5"`

Taking a `color(blue)"vector approach"`

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

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

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

`rArr4ulc=5ula'-ulb`

`color(white)(rArr4c)=5((-3),(-7))-((7),(2))`

`color(white)(rArr4c)=((-15),(-35))-((7),(2))`

`color(white)(rArr4c)=((-22),(-37))`

`rArrulc=1/4((-22),(-37))=((-11/2),(-37/4))`

`rArrC=(-11/2,-37/4)`