Question

1. 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:

To determine why all circles are similar, we use the definition of similar figures: figures that can be mapped onto each other by a sequence of translations, rotations, reflections, and dilations.

• Option A: Just stating the same shape is not a rigorous justification for similarity (similarity requires a specific transformation, not just same shape intuition).
• Option B: Lacking angles/sides does not imply similarity (e.g., a square and a rectangle both lack curves but aren’t always similar, and circles’ similarity is about transformations, not just the absence of angles/sides).
• Option C: Translating a circle onto another only works if they have the same radius (no dilation is mentioned, so it’s incorrect—different - radius circles can’t be translated onto each other without dilation).
• Option D: Translating the center to align them, then dilating (scaling) by an appropriate factor (to match radii) fits the similarity transformation definition. Dilations preserve shape (only scale size), so this shows circles can be mapped to each other via translation and dilation, proving similarity.

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.