Question

which transformation will not carry the rectangle below onto itself?
answer
a rotation of 180° clockwise about the origin
a reflection over the x - axis
a reflection over the line y = x
a reflection over the line y = -x

Explanation:

Step1: Analyze rotation of \(180^{\circ}\)

A rectangle has rotational symmetry of order \(2\). A \(180^{\circ}\) rotation about the center (in this case, the origin as the rectangle is symmetrically placed with respect to origin in terms of rotational symmetry) will map the rectangle onto itself.

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

For a rectangle (which is a parallelogram with right - angles), if we consider the transformation of reflection over the line \(y=x\). The opposite sides and angles of the rectangle are congruent. The reflection over \(y = x\) swaps the \(x\) and \(y\) coordinates of the vertices. Since the rectangle has a certain symmetry, this reflection will map the rectangle onto itself.

Step3: Analyze reflection over \(y=-x\)

The reflection over the line \(y =-x\) changes the coordinates \((x,y)\) to \((-y,-x)\). Due to the congruent opposite sides and angles of the rectangle, this reflection will map the rectangle onto itself.

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

The reflection over the \(x\) - axis changes the \(y\) - coordinate of each vertex \((x,y)\) to \((x, - y)\). If we assume the vertices of the rectangle (say \(A(x_1,y_1)\), \(B(x_2,y_2)\), \(C(x_3,y_3)\), \(D(x_4,y_4)\)). A non - square rectangle (as the given figure is a rectangle, not a square) does not have symmetry about the \(x\) - axis. The \(y\) - coordinates of the vertices above and below the \(x\) - axis are not in a symmetric relationship (in terms of reflection over \(x\) - axis for a non - square rectangle) to map the rectangle onto itself.

Answer:

a reflection over the \(x\) - axis