Question

a sequence of transformations maps triangle qrs onto congruent triangle xyz.
select all the sequences of transformations that could have mapped triangle qrs onto triangle xyz.
a reflection across the x - axis, followed by a 90 - degree clockwise rotation about the origin.
a 180 - degree clockwise rotation about the origin, followed by a reflection across the line y = 3.
a dilation by a scale factor of 3 centered at the origin, followed by a translation 5 units down.
a reflection across the line y = x, followed by a dilation by a scale factor of 1/3 centered at the origin.
a translation 1 unit left and 3 units up, followed by a 270 - degree counterclockwise rotation about the origin.

Explanation:

• Congruent triangles have the same shape and size. Transformations that preserve shape and size (isometries) include reflections, rotations, and translations. Dilations change the size (if scale factor ≠ 1) and thus do not map congruent triangles (unless scale factor = 1).
• For option 1: Reflection (isometry) and rotation (isometry) - both preserve shape and size.
• For option 2: Rotation (isometry) and reflection (isometry) - both preserve shape and size.
• For option 3: Dilation with scale factor 3 (changes size) - does not preserve congruence.
• For option 4: Dilation with scale factor \( \frac{1}{3} \) (changes size) - does not preserve congruence.
• For option 5: Translation (isometry) and rotation (isometry) - both preserve shape and size.

Answer:

A. a reflection across the x - axis, followed by a 90 - degree clockwise rotation about the origin.
B. a 180 - degree clockwise rotation about the origin, followed by a reflection across the line \( y = 3 \).
E. a translation 1 unit left and 3 units up, followed by a 270 - degree counterclockwise rotation about the origin.