Question

Points A and B are at `(3 ,7 )` and `(4 ,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

After Point A is rotated counterclockwise about the origin by `pi`, its new coordinates are `(-3, -7)`.

The difference between the `x` coordinates of Point A and B now is `4-(-3)=7` and `y` coordinates: `2-(-7)=9`

Since Point A was dilated about Point C by a factor of 5, we can find out by how much the coordinates change with each integer increase in factor.

For the `x` coordinate:
`7/4=1.75`
and `y` coordinate:
`9/4=2.25`

So for every integer increase in factor, the point moves 1.75 to the right and 2.25 upwards.

Point C is therefore
`(-3-1.75, -7-2.25)`
`=(-4.75, -9.25)`

Answer 2

`C=(-19/4,-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"`

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

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

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

`rArr4ulc=5ula'-ulb`

`color(white)(4ulcxx)=5((-3),(-7))-((4),(2))`

`color(white)(xxxx)=((-15),(-35))-((4),(2))=((-19),(-37))`

`rArrulc=1/4((-19),(-37))=((-19/4),(-37/4))`

`rArrC=(-19/4,-37/4)`