Question

Triangle A’B’C’ is the image of triangle ABC.

On a coordinate plane, triangle A B C has points negative 4, negative 1), (negative 5, negative 5), (negative 3, negative 4). Triangle A prime B prime C prime has points (3, 4), (4, 8), (2, 7).

Which transformations could have been used to create A’B’C’ ? Choose all that apply.
a 180degree rotation about the origin and then a translation 3 units up and 1 unit left
a translation 3 units up and 1 unit left and then a 180degree rotation about the origin
a 90degree clockwise rotation about the origin and then a reflection over the y-axis
a 90degree counterclockwise rotation about the origin and then a reflection over the x-axis
a translation 3 units down and 1 unit right and then a 180degree rotation about the origin

Answer 1

Correct answers: 1 and 5.

Reasoning (concise):
- 180° rotation about the origin: (x,y) -> (-x,-y).
- Then translate 3 up and 1 left: (-x,-y) -> (-x-1, -y+3).
Apply to the three original points:
(-4,-1) -> (4,1) -> (3,4)
(-5,-5) -> (5,5) -> (4,8)
(-3,-4) -> (3,4) -> (2,7)
So option 1 works.

- Option 5: translate 3 down and 1 right first: (x,y) -> (x+1, y-3), then 180° rotation -> (-(x+1), -(y-3)) = (-x-1, -y+3), which is the same net mapping as option 1. Applying it gives the same images above, so option 5 also works.

The other choices give different net transformations and do not map all three original vertices to the given A'B'C'.