Question

each sequence of transformations shown is applied to triangle lmn to create three different figures. select whether each sequence results in a figure that is similar, congruent, or both similar and congruent to triangle lmn.

• reflection across the x-axis, then a translation 7 units right
• (90^circ) counterclockwise rotation about the origin, then a translation 1 unit down
• dilation with a scale factor of (\frac{1}{3}) about the origin, then a reflection across the y-axis

(table with columns: similar, congruent (partially visible))

Explanation:

To solve this, we analyze each transformation sequence using the properties of congruence (preserves size and shape) and similarity (preserves shape, may change size):

1. Reflection across the \( x \)-axis, then translation 7 units right

• Reflection and translation are rigid transformations (they preserve side lengths and angles).
• Rigid transformations produce figures that are congruent (and thus also similar, since congruent figures are always similar).

2. \( 90^\circ \) counterclockwise rotation about the origin, then translation 1 unit down

• Rotation and translation are rigid transformations (preserve size and shape).
• Rigid transformations produce figures that are congruent (and thus also similar).

3. Dilation with scale factor \( \frac{1}{3} \), then reflection across the \( y \)-axis

• Dilation with scale factor \( \frac{1}{3} \) changes the size (scales down by \( \frac{1}{3} \)) but preserves shape.
• Reflection is a rigid transformation (preserves shape and size of the dilated figure).
• Together, these transformations produce a figure that is similar (shape preserved, size changed) but not congruent (size is not preserved).

Final Answers (Checkboxes):

Transformation Sequence	Similar	Congruent
Rotation + Translation	\( \boldsymbol{\checkmark} \)	\( \boldsymbol{\checkmark} \)
Dilation + Reflection	\( \boldsymbol{\checkmark} \)	\( \boldsymbol{\square} \)

(Note: For each row, check "Similar" and "Congruent" as per the analysis above.)

Answer:

To solve this, we analyze each transformation sequence using the properties of congruence (preserves size and shape) and similarity (preserves shape, may change size):

1. Reflection across the \( x \)-axis, then translation 7 units right

• Reflection and translation are rigid transformations (they preserve side lengths and angles).
• Rigid transformations produce figures that are congruent (and thus also similar, since congruent figures are always similar).

2. \( 90^\circ \) counterclockwise rotation about the origin, then translation 1 unit down

• Rotation and translation are rigid transformations (preserve size and shape).
• Rigid transformations produce figures that are congruent (and thus also similar).

3. Dilation with scale factor \( \frac{1}{3} \), then reflection across the \( y \)-axis

• Dilation with scale factor \( \frac{1}{3} \) changes the size (scales down by \( \frac{1}{3} \)) but preserves shape.
• Reflection is a rigid transformation (preserves shape and size of the dilated figure).
• Together, these transformations produce a figure that is similar (shape preserved, size changed) but not congruent (size is not preserved).

Final Answers (Checkboxes):

Transformation Sequence	Similar	Congruent
Rotation + Translation	\( \boldsymbol{\checkmark} \)	\( \boldsymbol{\checkmark} \)
Dilation + Reflection	\( \boldsymbol{\checkmark} \)	\( \boldsymbol{\square} \)

(Note: For each row, check "Similar" and "Congruent" as per the analysis above.)