Question

practice 2
which statement provides a valid justification for why all circles are similar?
a all circles have the same shape - a circle - so they must be similar.
b all circles have no angles and no sides, so they must be similar.
c i can translate any circle exactly onto another, so they must be similar.
d i can translate the center of any circle to the center of another, and then dilate from that center by an appropriate scale factor, so they must be similar.

Explanation:

Step1: Recall the definition of similar figures

Similar figures can be transformed into each other by a combination of translations, rotations, reflections, and dilations.

Step2: Analyze each option

• Option A: Just stating the same shape is too vague. Similarity has a precise transformation - based definition.
• Option B: Having no angles or sides is not a valid justification for similarity in the geometric transformation - based sense.
• Option C: Translation alone is not sufficient. A dilation is also needed if the circles have different sizes.
• Option D: Translating the center (a translation) and then dilating (a dilation) from that center by an appropriate scale factor is exactly the combination of transformations (translation and dilation) that can map one circle to another. This satisfies the definition of similar figures.

Answer:

D. I can translate the center of any circle to the center of another, and then dilate from that center by an appropriate scale factor, so they must be similar.