Triangle abc is transformed to create triangle def. what sequence of transformations maps one to the other? select all that apply.
a.
rotate 180° about the origin, then translate left 2 units
b.
reflect over the x-axis, then translate down 2 units
c.
translate 2 units to the right, then rotate 180° about the origin
d.
translate 2 units to the right, then reflect over the x-axis

B. Reflect over the x-axis, then translate down 2 units

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.

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