Question

drag and drop the answers into the boxes to correctly complete the statement. a sequence of transformations that maps △ def to △ def is a dashed box followed by a dashed box. options: translation 2 units right, rotation of 180° about the origin, reflection across the y-axis, translation 2 units up

Explanation:

To solve this, we analyze the transformation steps:

Step 1: Identify the first transformation

A \( 180^\circ \) rotation about the origin (or a reflection + translation, but rotation is a key step here) aligns the orientation. Alternatively, reflection across the \( y \)-axis could be first, but let's check the order. Wait, typically, a \( 180^\circ \) rotation (which is equivalent to \( (x,y) \to (-x,-y) \)) or reflection. Let's assume the first transformation is a rotation of \( 180^\circ \) about the origin (or reflection, but let's confirm the order). Then, a translation. Wait, the options are: rotation \( 180^\circ \), reflection \( y \)-axis, translation 2 right, translation 2 up.

Wait, let's think: To map \( \triangle DEF \) to \( \triangle D'E'F' \), first, a \( 180^\circ \) rotation (or reflection) followed by a translation. Let's check the coordinates (even without coordinates, the standard sequence: a \( 180^\circ \) rotation about the origin, then a translation 2 units up (or right)? Wait, no—wait, the options are:

First box: possible first transformation (rotation \( 180^\circ \) about origin, or reflection \( y \)-axis, or translation? No, translation first would not align orientation. So first, a rotation (or reflection) to align, then translation.

Wait, the correct sequence is: rotation of \( 180^\circ \) about the origin followed by translation 2 units up (or vice versa? Wait, no—wait, maybe reflection across \( y \)-axis? No, \( 180^\circ \) rotation is \( (x,y) \to (-x,-y) \), then translation 2 up: \( (-x, -y + 2) \). Alternatively, reflection across \( y \)-axis is \( (x,y) \to (-x,y) \), then translation 2 down? No, the options have translation 2 up.

Wait, the standard answer for such problems: A sequence is a rotation of \( 180^\circ \) about the origin followed by a translation 2 units up (or sometimes translation 2 right, but let's check the options). Wait, the options are:

First box: rotation of \( 180^\circ \) about the origin, reflection across \( y \)-axis, translation 2 right, translation 2 up.

Second box: the remaining.

Wait, let's confirm: A \( 180^\circ \) rotation about the origin changes \( (x,y) \) to \( (-x,-y) \). Then, translating 2 units up changes \( (-x,-y) \) to \( (-x, -y + 2) \), which would map \( \triangle DEF \) to \( \triangle D'E'F' \). Alternatively, reflection across \( y \)-axis ( \( (x,y) \to (-x,y) \)) followed by translation 2 down, but translation 2 down is not an option. So the correct first transformation is rotation of \( 180^\circ \) about the origin, followed by translation 2 units up.

Answer:

First box: rotation of \( 180^\circ \) about the origin
Second box: translation 2 units up

(Or, if the order is reversed, but typically rotation first. If the problem’s diagram shows \( \triangle DEF \) and \( \triangle D'E'F' \) with \( 180^\circ \) rotation then translation up, this is the sequence.)