Question

rectangle pqrs is transformed to create a second rectangle. both rectangles are shown in the coordinate grid below. which statements correctly explain why the two rectangles are congruent? select the two correct statements. a the second rectangle can be created by rotating pqrs 180° about the origin. b the second rectangle can be created by rotating pqrs 90° counterclockwise about the origin and then translating that figure 4 units to the right. c the second rectangle can be created by translating pqrs 2 units to the right and then translating that figure 2 units down. d the second rectangle can be created by rotating pqrs 90° clockwise about the origin and then reflecting that figure across the x - axis. e the second rectangle can be created by reflecting pqrs across the x - axis and then reflecting that figure across the y - axis.

Explanation:

Step1: Recall properties of congruent figures

Congruent figures have the same shape and size. Transformations that preserve shape and size are rotations, reflections, and translations.

Step2: Analyze each option

• Option A:

A \(180^{\circ}\) rotation about the origin is a rigid transformation. For a point \((x,y)\) rotated \(180^{\circ}\) about the origin, the new point is \((-x,-y)\). If we take the vertices of rectangle \(PQRS\) (say \(P(-1,2)\), \(Q(-5,2)\), \(R(-5,6)\), \(S(-1,6)\)) and rotate them \(180^{\circ}\) about the origin, we get new points \((1,-2)\), \((5,-2)\), \((5,-6)\), \((1,-6)\) which form a congruent rectangle.

• Option B:

A \(90^{\circ}\) counter - clockwise rotation about the origin: for a point \((x,y)\), the new point is \((-y,x)\). Then translating 4 units to the right (add 4 to the \(x\) - coordinate). This changes the position but also, if we consider the orientation and position changes, this combination does not map \(PQRS\) to the second rectangle.

• Option C:

Translating 2 units to the right (add 2 to \(x\) - coordinate) and 2 units down (subtract 2 from \(y\) - coordinate). If we take \(P(-1,2)\), after translation we get \((1,0)\) which is not a vertex of the second rectangle.

• Option D:

A \(90^{\circ}\) clockwise rotation about the origin: for a point \((x,y)\), the new point is \((y, - x)\). Then reflecting across the \(x\) - axis (change the sign of the \(y\) - coordinate). This combination of transformations does not map \(PQRS\) to the second rectangle.

• Option E:

Reflecting across the \(x\) - axis (change the sign of the \(y\) - coordinate) and then reflecting across the \(y\) - axis (change the sign of the \(x\) - coordinate). For a point \((x,y)\), after reflection across \(x\) - axis \((x,-y)\) and then across \(y\) - axis \((-x,-y)\) which is equivalent to a \(180^{\circ}\) rotation about the origin (a rigid transformation that preserves congruence).

Answer:

A. The second rectangle can be created by rotating \(PQRS\ 180^{\circ}\) about the origin.
E. The second rectangle can be created by reflecting \(PQRS\) across the \(x\) - axis and then reflecting that figure across the \(y\) - axis.