Question

which transformation will not carry the square below onto itself?
answer
a reflection over the line ( y = x )
a reflection over the line ( y = - 5 )
a rotation of ( 180 ^ { circ } ) counterclockwise about the origina reflection over the ( y )-axis

Explanation:

Step1: Analyze reflection over \(y = x\)

A square has symmetry about the line \(y=x\). Reflecting a square over \(y = x\) maps it onto itself.

Step2: Analyze reflection over \(y=-5\)

The square is centered around the origin (approximately from \(- 2\) to \(2\) on \(x\) - axis and \(-2\) to \(2\) on \(y\) - axis). The line \(y=-5\) is far below the square. Reflecting the square over \(y =-5\) will not map the square onto itself.

Step3: Analyze rotation of \(180^{\circ}\) about the origin

A square has \(180^{\circ}\) rotational symmetry about its center (the origin in this case). Rotating a square \(180^{\circ}\) about the origin maps it onto itself.

Step4: Analyze reflection over \(y\) - axis

A square has reflection symmetry about the \(y\) - axis. Reflecting a square over the \(y\) - axis maps it onto itself.

Answer:

a reflection over the line \(y=-5\)