Question

Points A and B are at `(4 ,5 )` and `(7 ,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

`C(9,8)`

Explanation:

Given: `A(4,5) and B(7,2)` A is rotated by `3pi/2` and dilated by a factor of 1/2 about a point C. After rotation and dilation A's new location is on B, that is `A^(RD)(7,2)`.
Required: the coordinates of C such that `A^(RD)(7,2) = B(7,2)`
Solution Strategy: a) Rotate `A`, by `R(3/2pi)`
b) Using Dilation about a `C` construct and knowing `A^(RD)` solve C
a) Rotation of `A` by `3/2pi`
`A^R = R(3/2pi) A`
`R(3/2pi)=[(cos(3/2pi),-sin(3/2pi) ),(sin(3/2pi),cos(3/2pi))]= [(0,1),(-1,0)]`
`A^R=[(0,1),(-1,0)] [(4),(5)] =[(5),(-4)]`
b) In order to dilate about C(x,y) we need to do do the following:
i) Translation `A^R` by `C(x,y)`
ii) Dilate by 1/2,
iii) Undo the translation:
Putting i), ii) and iii) we can wrtite:
`A^(RD) =[(7),(2)] = [(1/2(5-x)+x), (1/2(-4-y)+y ) ]` solve for `x and y`
`7=5/2-x/2+x; x=9`
`2=-2-y/2+y; y=8`
`C(9,8)`