Question

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

Explanation:

Step1: Analyze rotation about \(P\) by \(180^{\circ}\)

A rectangle has rotational symmetry of order \(2\) about its center. A \(180^{\circ}\) rotation about the center \(P\) of a rectangle maps each vertex \((x,y)\) to \((-x,-y)\) (in a coordinate - system with \(P\) as the origin). Since the opposite sides of a rectangle are equal and parallel, a counter - clockwise rotation about \(P\) by \(180^{\circ}\) maps the rectangle onto itself.

Step2: Analyze rotation about \(P\) by \(90^{\circ}\)

A rectangle (unless it is a square) does not have \(90^{\circ}\) rotational symmetry. For a non - square rectangle, after a \(90^{\circ}\) clockwise rotation about the center \(P\), the sides of the rectangle (which have different lengths) will not map onto their original positions.

Step3: Analyze reflection across line \(a\)

Line \(a\) is a diagonal. For a rectangle (non - square), the two triangles formed by a diagonal are congruent, but the reflection across the diagonal does not map the rectangle onto itself. Because the adjacent sides of a non - square rectangle are not equal, the reflection across the diagonal will not map each side onto itself.

Step4: Analyze reflection across line \(b\)

Line \(b\) is a line that bisects the sides of the rectangle. A rectangle has reflection symmetry across the lines that bisect its opposite sides. When we reflect a rectangle across a line that bisects its opposite sides (in this case, line \(b\)), each point on one side of the line has a corresponding point on the other side of the line such that the line is the perpendicular bisector of the segment joining the two points. So, reflection across line \(b\) maps the rectangle onto itself.

Answer:

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