The point G(5,-9) is rotated 90 degrees about point M(-8,3) and then reflected across the line y=9. Find the coordinates of the image G'.

Question

I have no idea where to start. Please advice.

Answer 1

suppose you rotate around (0,0). That takes (x,y) -> (-y,x)

Shifting the origin to (-8,3) changes the distance, and gives us (x+8,y-3). So, the shift takes

(5,-9) -> (13,-12)
Then the rotation takes
(13,-12) -> (12,13)

reflection through the line y=9 takes (x,y) -> (x,9-(y-9)) = (x,18-y)
Thus we end up with (12,5)

Answer 2

bye this not even an answer choice

Answer 3

Let's switch coordinates and look at G using M as (0,0).
From M to G, the x distance is 5-(-8)=13
and the y distance is -9-3=-12
So in M coordinates, G is (13,-12).
To rotate about M by 90 degrees then G becomes (12,13) or (-12,-13) depending on positive or negative rotation by 90, since you didn't specify.
Remember these are in M coordinates.
So to change back to the original coordinates, we have to add back the coordinates of M.


(12,13)+(-8,3)=(4,16)
(-12,-13)+(-8,3)=(-20,-10)
So now if we reflect about y=9, you find the y-distance from y=9 and then add that distance to 9 to get the new y-coordinate.
So for (4,16), the distance from y=9 is 16-9=7.
Since the point is above y=9, we will subtract 7 from the y=9 to get the reflected point.
(4,9-7)=(4,2)
and for (-20,-10), the distance to y=9 is 9-(-10)=19 so then add 19 to y=9
(-20,9+19)=(-20,28)
So then G' is either (4,2) or (-20,28).

Hope this helps!