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'.

I have no idea where to start. Please advice.

bye this not even an answer choice

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!

To find the coordinates of the image G' after the given transformations, we can follow these steps:

1. Start by performing a 90-degree counterclockwise rotation about point M(-8,3).
- To rotate a point (x, y) counterclockwise by 90 degrees about point M(x₀, y₀), we can use the following formulas:
x' = x₀ + (y - y₀)
y' = y₀ - (x - x₀)

Let's substitute the values:
x' = -8 + (-9 - 3) = -8 + (-12) = -20
y' = 3 - (5 - (-8)) = 3 - 13 = -10

After rotation, point G(5, -9) becomes G₁(-20, -10).

2. Next, we need to reflect point G₁ across the line y = 9.
- Reflection across the line y = 9 can be done by finding the distance between a point and the reflection line and then subtracting it from the y-coordinate of the reflection line.

The line y = 9 is horizontal and has a distance of 9 units from the origin. Thus, to reflect a point across this line, we subtract twice the distance between the point and the reflection line from the y-coordinate.

Let's calculate:
distance = 9 - (-10) = 9 + 10 = 19
y" = 9 - 2 * 19 = 9 - 38 = -29

After reflection, point G₁(-20, -10) becomes G'(-20, -29).

Therefore, the coordinates of the image G' are (-20, -29).

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)