Question

a rectangle is shown below.
line a intersects two vertices.
line b bisects each side it passes through.
point k is the center of the rectangle.
which transformation(s) must map the rectangle exactly onto itself? choose all that apply.
reflection across line a
clockwise rotation about k by 180°
counterclockwise rotation about k by 90°
reflection across line b
none of the above

Explanation:

Step1: Analyze reflection across line \(a\)

A rectangle is not symmetric about a diagonal (line \(a\)). When we reflect a rectangle across its diagonal, the image does not coincide with the original rectangle.

Step2: Analyze clockwise rotation about \(K\) by \(180^{\circ}\)

For a rectangle, a \(180^{\circ}\) rotation about its center \(K\) maps each vertex \((x,y)\) to \((-x,-y)\) (in a coordinate system with \(K\) as the origin). Since a rectangle has opposite sides equal and parallel, a \(180^{\circ}\) rotation about its center maps the rectangle onto itself.

Step3: Analyze counter - clockwise rotation about \(K\) by \(90^{\circ}\)

A rectangle (unless it is a square) does not have \(90^{\circ}\) rotational symmetry. After a \(90^{\circ}\) rotation about its center \(K\), the sides of the rectangle will not map onto their original positions.

Step4: Analyze reflection across line \(b\)

Line \(b\) is a line that bisects two opposite sides of the rectangle. A rectangle is symmetric about the line that bisects its opposite sides. When we reflect a rectangle across the line that bisects its opposite sides (line \(b\)), the image coincides with the original rectangle.

Answer:

Clockwise rotation about \(K\) by \(180^{\circ}\), Reflection across line \(b\)