Question

1. 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. a reflection across the x-axis, followed by a 90° clockwise rotation about the origin b. a 180° clockwise rotations about the origin, followed by a reflection across the line y = 3 c. a dilation by a scale factor of 3 centered at the origin, followed by a translation 5 units down d. a reflection across the line y = x, followed by a dilation by a scale factor of \\(\frac{1}{3}\\) centered at the origin e. a translation 1 unit left and 3 units up, followed by a 270° counterclockwise rotation about the origin

Explanation:

Step1: Recall Congruence Transformations

Congruent figures can be mapped using rigid transformations (reflection, rotation, translation) which preserve size and shape. Dilations are non - rigid (change size) unless scale factor is 1, so options with dilation (c, d) are out as they change size (scale factor ≠1), so c and d are eliminated.

Step2: Analyze Option a

Reflection across x - axis (rigid) and 90° clockwise rotation (rigid) are both rigid transformations. So this sequence can map QRS to XYZ (congruent).

Step3: Analyze Option b

180° clockwise rotation (rigid) and reflection across \(y = 3\) (rigid) are rigid transformations. So this sequence can map QRS to XYZ (congruent).

Step4: Analyze Option e

Translation (1 unit left, 3 units up - rigid) and 270° counter - clockwise rotation (rigid) are rigid transformations. So this sequence can map QRS to XYZ (congruent).

Answer:

a. a reflection across the x - axis, followed by a \(90^{\circ}\) clockwise rotation about the origin
b. a \(180^{\circ}\) clockwise rotations 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 \(270^{\circ}\) counterclockwise rotation about the origin