Triangle A'B'C' is the image of triangle ABC Which transformations could have been used to create ABC? Choose all that apply

Point A, initially at (-4, -1), moves to A' at (3, 4).

Point B (-5, -5) moves to B' (4, 8).

Point C (-3, -4) moves to C' (2, 7).

The first choice: rotating 180 degrees about the origin changes to sign of both x- and y-coordinates, so A(-4, -1) goes to (4, 1). Moving 3 up goes to (4, 4), and 1 left is (3, 4). For B(-5, -5), rotating brings it to (5, 5), up 3 to (5, 8), 1 left to (4, 8). For C(-3, -4), rotate to (3, 4), up 3 to (3, 7), 1 left to (2, 7). So since all points match up, this first choice is a correct answer.

The second choice does not work since: translating A 3 up and 1 left brings it to (-5, 2), and rotating it 180 degrees would then put A' at (5, -2).

The third choice: rotating A 90 degrees clockwise puts it at (-1, 4), and reflecting it over the y-axis puts it at (1, 4).

The fourth choice: rotation A 90 degrees counterclockwise puts it at (1, -4), and reflecting it over the x-axis puts it at (1, 4).

The fifth choice: translating 3 down and 1 right puts A at (-3, -4), then a 180-degree rotation puts it at (3, 4). B(-5, -5) translated goes to (-4, -8), and rotating brings it to  (4, 8). C(-3, -4) is translated to (-2, -7), and rotating puts it at (2, 7). So since all points match up, this choice is another correct answer.

So the possible transformations are the 1st and 5th choices.

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