Question

1. a student claims: \if you rotate a rectangle 180° about its center and then dilate it by a scale factor of ½, the image will be congruent to the original rectangle.\

which statement best explains whether the student is correct?

the student is correct, because rotations and dilations always produce congruent figures.

the student is incorrect, because a dilation with scale factor ½ changes the side lengths.

the student is incorrect, because rotations do not preserve shape.

the student is correct, because both rotation and dilation preserve congruence.

Explanation:

1. Recall the definitions of rotation and dilation:

• Rotation is a rigid transformation that preserves side lengths and angles, so it preserves congruence.
• Dilation is a non - rigid transformation that changes the size of the figure (scales side lengths by the scale factor) and does not preserve congruence unless the scale factor is 1.

1. Analyze the student's claim: The student applies a rotation (which preserves congruence) and then a dilation with scale factor \( \frac{1}{2} \) (which changes side lengths). For two figures to be congruent, their corresponding side lengths must be equal. Since the dilation changes the side lengths, the image after dilation will not be congruent to the original rectangle.

• Option 1: Dilations do not always produce congruent figures (only when scale factor is 1), so this is wrong.
• Option 2: A dilation with scale factor \( \frac{1}{2} \) will halve the side lengths of the rectangle. Since congruent figures must have equal corresponding side lengths, the image after dilation will not be congruent to the original. So this option correctly explains the student is incorrect.
• Option 3: Rotations are rigid transformations and do preserve shape, so this is wrong.
• Option 4: Dilation does not preserve congruence (except when scale factor = 1), so this is wrong.

Answer:

The student is incorrect, because a dilation with scale factor ½ changes the side lengths.