Question

which transformation will not carry the square below onto itself?
answer
a reflection over the line $y = x$
a reflection over the line $y = -x$
a rotation of $90^\circ$ counterclockwise about the origin
a reflection over the line $y = -x + 5$

Explanation:

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

A square has symmetry about the line \(y=x\). When reflected over \(y = x\), the square maps onto itself.

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

A square has symmetry about the line \(y =-x\). When reflected over \(y=-x\), the square maps onto itself.

Step3: Analyze rotation of \(90^{\circ}\) counter - clockwise about the origin

A square has rotational symmetry of order \(4\) (i.e., \(90^{\circ},180^{\circ},270^{\circ},360^{\circ}\) rotations about the center). Rotating \(90^{\circ}\) counter - clockwise about the origin (the center of the square) maps the square onto itself.

Step4: Analyze reflection over \(y=-x + 5\)

The line \(y=-x + 5\) is not a line of symmetry for the square. When we reflect the square over the line \(y=-x + 5\), the square will not map onto itself.

Answer:

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