Answer:
B. Reflect over the x-axis, then translate down 2 units
Explanation:
Since we are dealing with rigid transformations (no dilation), both triangles are congruent, which means if point A maps to D and D maps to A after the sequence of transformation, the rest of the points will map as well. So, to prove that a sequence of transformations maps one triangle to the other, we just need to prove A = D and D = A after the transformations.
We can infer from the picture that A = (-4, 2) and D = (-4, -4), so let's apply each of the sequences:
A. - The rule to reflect a point 180° about the origin is (x, y) → (-x, -y). In other words, we change the sign of both coordinates.
- The rule to translate a point 2 units left is (x, y) → (x-2, y). In other words, we subtract 2 from the x-coordinate.
∴ Rotate A 180° about the origin: (x, y) → (-x, -y) = (-4, 2) → (4, -2)
∴ Translate the resulting point 2 units left: (x, y) → (x-2, y) = (4-2, -2) = (2, -2)
The resulting point is not equal D (2, -2) ≠ (-4, -4), so this is not one of our sequences.
B. - The rule to reflect a point over the x-axis is (x, y) → (x, -y). In other words, we change the signs of the y-coordinate.
- The rule to translate a point 2 units down is (x, y) → (x, y-2). We subtract two from the y-coordinate.
∴ Reflect A over the x-axis: (x, y) → (x, -y) = (-4, -2)
∴ Translate the resulting point 2 units down: (x, y) → (x, y-2) = (-4, -2-2) = (-4, -4)
Notice that we end up at D, so lets check if the opposite holds:
∴ Reflect D over the x-axis: (x, y) → (x, -y) =(-4, -4) → (-4, 4)
∴ Translate the resulting point 2 units down: (x, y) → (x, y-2) = (-4, 4-2) = (4, 2)
We end up at A, so after the sequence of transformations A = D and D = A. We can conclude that this sequence of transformations maps one triangle to the other.
C. - The rule to translate a point 2 units right is (x, y) → (x+2, y). In other words, we add 2 from the x-coordinate.
- The rule to reflect a point 180° about the origin is (x, y) → (-x, -y).
∴ Translate A 2 units right: (x, y) → (x+2, y) = (-4+2, 2) = (-2, 2)
∴ Rotate the resulting point 180° about the origin: (x, y) → (-x, -y) = (-2, 2) → (2 , -2)
The resulting point is not equal D (2, -2) ≠ (-4, -4), so this is not one of our sequences.
D. - The rule to translate a point 2 units right is (x, y) → (x+2, y).
- The rule to reflect a point over the x-axis is (x, y) → (x, -y).
∴ Translate A 2 units right: (x, y) → (x+2, y) = (-4+2, 2) = (-2, 2)
∴ Reflect the resulting point over the x-axis: (x, y) → (x, -y) = (-2, 2) → (-2 , -2)
The resulting point is not equal D (-2, -2) ≠ (-4, -4), so this is not one of our sequences.