Answer:
- reflection over the y-axis
- translation left 4, down 3
Explanation:
You want two transformations that will map segment AB to segment A"B", where the points are A(1, 1), B(5, 3), A"(-5, -2), B"(-9, 0).
Reflection
We notice that the difference B-A is (5 -1, 3 -1) = (4, 2), and the difference B"-A" is (-9 -(-5), 0 -(-2)) = (-4, 2). These values have the same magnitudes, so we know no dilation is involved. The sign of the x direction of the segment has been reversed, which means the segment has been reflected horizontally. (B" is left of A", where B is right of A.)
Reflection over any vertical line will do, but the simplest is reflection over the y-axis.
The first transformation can be reflection over the y-axis.
Translation
The reflected line segment will be ...
(x, y) ⇒ (-x, y)
A(1, 1) ⇒ A'(-1, 1)
B(5, 3) ⇒ B'(-5, 3)
The value that must be added to A' to move it to A" is ...
A" -A' = (-5, -2) -(-1, 1) = (-5 -(-1), -2-1) = (-4, -3) . . . . . . left 4, down 3
Similarly, the value that must be added to B' to move it to B" is ...
B" -B' = (-9, 0) -(-5, 3) = (-9 -(-5), 0 -3) = (-4, -3) . . . . . . left 4, down 3
That is, the entire segment A'B' can be moved to A"B" by a translation left 4 and down 3.
The second transformation can be translation left 4 and down 3.
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Additional comment
The translation left could be accomplished by reflection over the line x = -2 instead of the y-axis.
The movement of the line segment could be accomplished in one transformation by rotation 126.87° clockwise about the point (-1.25, -2). The exact angle is arctan(-4/3) in the second quadrant.