Question

which transformations were used to create congruent trapezoid wxyz from trapezoid wxyz?
translation and rotation of 180° about the origin
translation and reflection across the y - axis
rotation of 180° about the origin and a reflection across the y - axis
rotation of 90° clockwise about the origin and a reflection across the x - axis

Explanation:

Step1: Analyze Rotation 180°

A 180° rotation about the origin changes \((x,y)\) to \((-x,-y)\). Translating first, then rotating 180°: Let's check coordinates. Original (orange) and final (green) – rotation 180° flips signs, translation moves it. But let's check reflection/rotation.

Step2: Analyze Reflection and Translation

Reflection over y - axis changes \((x,y)\) to \((-x,y)\). But the green trapezoid is in lower right, orange in upper left. A 180° rotation (swaps x and y signs) and translation? Wait, the first option: translation (moving) and 180° rotation about origin. Let's see: 180° rotation flips both x and y signs, then translation. Alternatively, check the congruence (same shape, size). Rotation 180° and translation would map the orange to green. Wait, the correct option: Let's re - evaluate. The orange is in second quadrant (x negative, y positive), green in fourth (x positive, y negative). A 180° rotation about origin takes (x,y) to (-x,-y), but if we translate first? Wait, no. Wait, the first option: "translation and rotation of 180° about the origin". Let's think of a point in orange: say W' has coordinates (let's assume grid: each square is 1 unit). Suppose W' is at (-6,6), after 180° rotation: (6,-6), then translation? Wait, no, maybe the other way. Wait, the green trapezoid is in (positive x, negative y), orange in (negative x, positive y). A 180° rotation about origin (which is (0,0)) would take (x,y) to (-x,-y), but if we translate first, then rotate. Alternatively, the correct answer is "translation and rotation of 180° about the origin". Wait, no, let's check the options again. Wait, the first option is "translation and rotation of 180° about the origin", second is "translation and reflection across the y - axis", third is "rotation of 180° about the origin and a reflection across the y - axis", fourth is "rotation of 90° clockwise about the origin and a reflection across the x - axis". Let's analyze the orientation. The orange trapezoid and green trapezoid: a 180° rotation would flip both x and y signs, and translation. So the first option is correct.

Answer:

A. translation and rotation of \(180^{\circ}\) about the origin