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:

• Option A: Rotation and translation are rigid transformations. A rotation of \(90^{\circ}\) (which is a rigid transformation that preserves shape and size) followed by a translation (another rigid transformation that preserves shape and size) will result in a congruent image.
• Option B: Reflection (across the \(x -\)axis, a rigid transformation) and rotation (\(270^{\circ}\), a rigid transformation) are both rigid transformations. Rigid transformations preserve the shape and size of the figure, so the resulting image is congruent to the original.
• Option C: Dilation is a non - rigid transformation. A dilation by a scale factor of \(3\) changes the size of the figure. Even though translation (a rigid transformation) is involved first, the dilation makes the image similar (same shape, different size) but not congruent.
• Option D: Rotation (\(180^{\circ}\), a rigid transformation) and reflection (across the \(y -\)axis, a rigid transformation) are both rigid transformations. Rigid transformations preserve the shape and size of the figure, so the resulting image is congruent to the original.

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.