Math Snap

STEP 1

Assumptions1. The triangles $\triangle QRS$ and \triangleUV are congruent.
. The transformations are performed on $\triangle QRS$.
3. The transformations include reflection across the $x$-axis, reflection across the $y$-axis, rotation around the origin, and translation to the left.
4. The transformations can be performed in any sequence.

STEP 2

We need to determine which sequences of transformations will result in a triangle congruent to \triangleUV.

STEP 3

For option A, reflect $\triangle QRS$ across the $x$-axis. Then reflect the image across the $y$-axis. Then reflect the image across the $x$-axis.

STEP 4

Reflecting a figure across the $x$-axis changes the sign of the $y$-coordinates. Reflecting the figure again across the $y$-axis changes the sign of the $x$-coordinates. Reflecting the figure a third time across the $x$-axis changes the sign of the $y$-coordinates again.

STEP 5

The net effect of these three reflections is equivalent to a reflection across the $y$-axis. Therefore, option A is a valid sequence of transformations.

STEP 6

For option B, reflect $\triangle QRS$ across the $y$-axis.

STEP 7

Reflecting a figure across the $y$-axis changes the sign of the $x$-coordinates. This transformation alone can result in a triangle congruent to \triangleUV. Therefore, option B is a valid sequence of transformations.

STEP 8

For option C, rotate $\triangle QRS$ $90^{\circ}$ clockwise around the origin. Then reflect the image across the $y$-axis.

STEP 9

Rotating a figure $90^{\circ}$ clockwise around the origin swaps the $x$ and $y$-coordinates and changes the sign of the new $y$-coordinates. Reflecting the figure across the $y$-axis changes the sign of the $x$-coordinates.

STEP 10

The net effect of these two transformations is equivalent to a rotation of $90^{\circ}$ counterclockwise around the origin. Therefore, option C is not a valid sequence of transformations.

STEP 11

For option D, rotate $\triangle QRS$ $180^{\circ}$ around the origin. Then reflect the image across the $x$-axis.

STEP 12

Rotating a figure $180^{\circ}$ around the origin changes the sign of both the $x$ and $y$-coordinates. Reflecting the figure across the $x$-axis changes the sign of the $y$-coordinates.

STEP 13

The net effect of these two transformations is equivalent to a reflection across the $y$-axis. Therefore, option D is a valid sequence of transformations.

STEP 14

For option, translate $\triangle QRS$ $4$ units to the left.

Translating a figure does not change the shape or size of the figure, only its position. Therefore, option is not a valid sequence of transformations.
The correct sequences of transformations are A, B, and D.