Question

which series of transformations demonstrates that the figures are congruent?
a a reflection over the x - axis followed by a reflection over the y - axis
b a translation 6 units up followed by a rotation of 180° counterclockwise about the origin
c a translation 3 units up followed by a reflection over the y - axis
d a rotation of 270° counterclockwise about the origin followed by a translation over the y - axis

Explanation:

Step1: Analyze Option A

A reflection over the \(x\) - axis followed by a reflection over the \(y\) - axis.
Let \((x,y)\) be a point. Reflection over \(x\) - axis gives \((x, - y)\). Reflection over \(y\) - axis gives \((-x,-y)\).
If we consider two congruent figures (which are mirror - images in both axes), this combination of reflections can map one figure to the other.

Step2: Analyze Option B

A translation \(6\) units up followed by a rotation of \(90^{\circ}\) counter - clockwise about the origin.
A translation \(6\) units up changes the \(y\) - coordinate of each point by \(+6\). A \(90^{\circ}\) counter - clockwise rotation about the origin transforms a point \((x,y)\) to \((-y,x)\). This combination will not map the given figures (by visual inspection of the grid - based figures, the orientation and position changes do not match the congruent figure mapping).

Step3: Analyze Option C

A translation \(3\) units up followed by a reflection over the \(y\) - axis.
A translation \(3\) units up changes the \(y\) - coordinate of each point by \(+3\). A reflection over the \(y\) - axis changes a point \((x,y)\) to \((-x,y)\). By visual inspection of the grid, this combination will not map the two figures (the vertical shift and reflection do not align the figures).

Step4: Analyze Option D

A rotation of \(270^{\circ}\) counter - clockwise about the origin followed by a translation over the \(y\) - axis.
A \(270^{\circ}\) counter - clockwise rotation about the origin transforms a point \((x,y)\) to \((y, - x)\). A translation over the \(y\) - axis is not a standard transformation (translation is in terms of horizontal or vertical shifts, not "over the \(y\) - axis" in the correct transformation sense). Also, by visual inspection of the grid, this combination will not map the two figures.

Answer:

A. A reflection over the \(x\) axis followed by a reflection over the \(y\) axis.