Question

which of the following composition of transformations would create an image that is not congruent to its original image?

a rotation of 45° followed by a reflection across the x - axis

a translation 2 units to the left followed by dilation by 1/2

a reflection across the y - axis followed by a rotation of 60°

a rotation of 135° followed by a translation of 4 units to the right

Explanation:

Step1: Recall congruence transformations

Congruence transformations (isometries) are translation, rotation, reflection. These preserve size and shape, so the image is congruent to the original. Dilation is a non - congruence transformation as it changes the size (scales the figure) unless the scale factor is 1.

Step2: Analyze each option

• Option 1: Rotation (isometry) followed by reflection (isometry). The composition of isometries is an isometry, so the image is congruent.
• Option 2: Translation (isometry) followed by dilation (non - isometry with scale factor \( \frac{1}{2}\)). Dilation changes the size, so the image is not congruent to the original.
• Option 3: Reflection (isometry) followed by rotation (isometry). The composition of isometries is an isometry, so the image is congruent.
• Option 4: Rotation (isometry) followed by translation (isometry). The composition of isometries is an isometry, so the image is congruent.

Answer:

B. A translation 2 units to the left followed by dilation by \( \frac{1}{2}\)