Question

question: a sequence of transformations is applied to △abc to create △abc. select all sequences that could result in △abc = △abc (congruent triangles). a) translation 5 units right and 3 units up, then a 90° clockwise rotation about the image of a. b) reflection across the line y = x, then a translation 4 units left. c) dilation about point a with scale factor 2. d) 180° clockwise rotation about the origin. e) dilation about the origin with scale factor 1, then a translation 2 units down. f) reflection across the xxx - axis, then a dilation with scale factor 0.5.

Explanation:

Step1: Recall congruence - preserving transformations

Rigid motions (translations, rotations, reflections) preserve congruence. Dilations with a scale - factor other than 1 change the size of the triangle and thus do not preserve congruence.

Step2: Analyze Option A

Translation and rotation are rigid motions. But a 90 - degree clockwise rotation changes the orientation. However, combined with translation, it is a sequence of rigid motions, so it can result in congruent triangles.

Step3: Analyze Option B

Reflection and translation are rigid motions. A reflection across y = x and then a translation will result in a congruent triangle.

Step4: Analyze Option C

Dilation with a scale factor of 2 changes the size of the triangle. So, $\triangle ABC$ and $\triangle A'B'C'$ will not be congruent.

Step5: Analyze Option D

A 180 - degree clockwise rotation about the origin is a rigid motion and will result in a congruent triangle.

Step6: Analyze Option E

Dilation with a scale factor of 1 is equivalent to no dilation (since multiplying the side - lengths by 1 does not change them), and then a translation is a rigid motion. So, it can result in congruent triangles.

Step7: Analyze Option F

Dilation with a scale factor of 0.5 changes the size of the triangle. So, $\triangle ABC$ and $\triangle A'B'C'$ will not be congruent.

Answer:

A. Translation 5 units right and 3 units up, then a 90° clockwise rotation about the image of A.
B. Reflection across the line y = x, then a translation 4 units left.
D. 180° clockwise rotation about the origin.
E. Dilation about the origin with scale factor 1, then a translation 2 units down.