Question

which combinations of transformations will always produce congruent figures? check all that apply.
□ a dilation, followed by a rotation
□ a reflection, followed by a dilation
□ a rotation, followed by a translation
□ a translation, then a rotation, followed by a stretch
□ a translation, followed by two reflections

Explanation:

Step1: Recall congruence transformations

Congruence transformations (isometries) are transformations that preserve the shape and size of a figure. These include translations, rotations, and reflections. Dilations and stretches are not isometries as they change the size (dilation scales, stretch deforms/stretches).

Step2: Analyze each option

• Option 1: dilation, followed by rotation

Dilation changes the size (scales the figure), so the resulting figure is not congruent to the original. Eliminate this.

• Option 2: reflection, followed by dilation

Dilation changes the size, so the result is not congruent. Eliminate this.

• Option 3: rotation, followed by translation

Rotation and translation are both isometries (preserve size/shape). Their combination preserves congruence. Keep this.

• Option 4: translation, then rotation, followed by stretch

Stretch changes the size/shape, so the result is not congruent. Eliminate this.

• Option 5: translation, followed by two reflections

Translation and reflections are isometries. Their combination preserves congruence. Keep this.

Answer:

C. a rotation, followed by a translation
E. a translation, followed by two reflections

(Note: Using the option labels as per the original (though the original uses checkboxes, we identify the correct ones by their descriptions):

• "a rotation, followed by a translation"
• "a translation, followed by two reflections" )