Question

which of the following describes the transformation from △abc→△abc? reflection over x - axis, then translation (0,2) 90° counterclockwise rotation, then reflection over y - axis reflection over y - axis, then rotation 180° 90° counterclockwise rotation, then reflection over x - axis

Explanation:

Step1: Analyze reflection over x - axis

If we reflect a point $(x,y)$ over the x - axis, the transformation is $(x,y)\to(x, - y)$. If we first reflect $\triangle ABC$ over the x - axis, the orientation of the triangle will be upside - down. But the orientation of $\triangle A'B'C'$ is not upside - down compared to $\triangle ABC$, so the first option is incorrect.

Step2: Analyze 90° counterclockwise rotation

The rule for a 90° counterclockwise rotation about the origin is $(x,y)\to(-y,x)$. If we first perform a 90° counterclockwise rotation on $\triangle ABC$ and then reflect over the y - axis (rule $(x,y)\to(-x,y)$), the resulting triangle will not match $\triangle A'B'C'$ in position and orientation, so the second option is incorrect.

Step3: Analyze reflection over y - axis and 180° rotation

The rule for reflection over the y - axis is $(x,y)\to(-x,y)$. After reflection over the y - axis, and then a 180° rotation (rule $(x,y)\to(-x,-y)$), the resulting triangle will not match $\triangle A'B'C'$, so the third option is incorrect.

Step4: Analyze 90° counterclockwise rotation and reflection over x - axis

First, for a 90° counterclockwise rotation about the origin, the rule is $(x,y)\to(-y,x)$. Then, for reflection over the x - axis, the rule is $(x,y)\to(x, - y)$. If we apply these two transformations successively to the vertices of $\triangle ABC$, we can get $\triangle A'B'C'$.

Answer:

90° counterclockwise rotation, then reflection over x - axis