Question

choose all of the transformation sequences which create an image congruent to the original image.
a a rotation of 90° followed by a translation of 3 units to the right.
b a reflection across the x - axis followed by a rotation of 270°.
c a translation 6 units to the left followed by a dilation by 3.
d a rotation of 180° followed by a reflection across the y - axis.

Explanation:

Step1: Recall the properties of congruent transformations

Congruent transformations include translations, rotations, and reflections. Dilations change the size of a figure, so any transformation sequence with a dilation will not result in a congruent image.

Step2: Analyze each option

• Option A: A rotation of \(90^{\circ}\) followed by a translation of 3 units to the right. Rotation and translation are congruent transformations.
• Option B: A reflection across the \(x -\)axis followed by a rotation of \(270^{\circ}\). Reflection and rotation are congruent transformations.
• Option C: A translation 6 units to the left followed by a dilation by 3. Dilation is not a congruent transformation.
• Option D: A rotation of \(180^{\circ}\) followed by a reflection across the \(y -\)axis. Rotation and reflection are congruent transformations.

Answer:

A. A rotation of \(90^{\circ}\) followed by a translation of 3 units to the right, B. A reflection across the \(x -\)axis followed by a rotation of \(270^{\circ}\), D. A rotation of \(180^{\circ}\) followed by a reflection across the \(y -\)axis.