Find the image of O (-2, -1) after two reflections, first across the line y=-5, and then across the line x=1.

(-2, -1)
(-1, -6)
( 4, -9)
( 1, -5)

I honestly think that the correct answer is not listed. Please help!

(-2,-1) -> (-2,-9) -> (4,-9)

To find the image of the point O (-2, -1) after two reflections, first across the line y = -5 and then across the line x = 1, we can follow these steps:

1. Reflection across the line y = -5:
- The line y = -5 acts as the mirror line.
- To reflect a point across this line, we can calculate the distance between the point and the mirror line, and then move the point the same distance on the other side of the line.
- The distance between the point O (-2, -1) and the mirror line y = -5 is |-1 - (-5)| = 4.
- Therefore, the point O (-2, -1) reflects to the point O' (-2, -9).

2. Reflection across the line x = 1:
- The line x = 1 acts as the mirror line.
- To reflect a point across this line, we can calculate the distance between the point and the mirror line, and then move the point the same distance on the other side of the line.
- The distance between the point O' (-2, -9) and the mirror line x = 1 is |-2 - 1| = 3.
- Therefore, the point O' (-2, -9) reflects to the point O'' (3, -9).

So, the image of O (-2, -1) after two reflections is O'' (3, -9).

Therefore, none of the given options ((-2, -1), (-1, -6), (4, -9), (1, -5)) is the correct answer. The correct answer is (3, -9).

To find the image of a point after two reflections, we can apply the following steps:

1. Reflect the point across the line y = -5:
- The line y = -5 acts as the mirror line. To find the image of point O(-2, -1) after this reflection, we need to find the point which is equidistant from the mirror line as O.
- As O is 4 units above the mirror line, its image will be 4 units below the mirror line.
- Therefore, the image after the first reflection is O'(-2, -9).

2. Reflect the newly obtained point across the line x = 1:
- The line x = 1 acts as the new mirror line. To find the final image, we need to find the point equidistant from the mirror line as O'.
- As O' is 3 units to the left of the mirror line, its image will be 3 units to the right of the mirror line.
- Therefore, the image after the second reflection is O''(1, -9).

The answer that matches this process is (1, -9) (option D). Hence, the correct answer is (1, -9).