Geometry Question of the Day

1. They are similar because a dilation centered at $O$ with scale factor $\frac{10}{4}$ maps the smaller circle to the larger circle. (correct answer)
2. They are similar because they share the same center, so they are congruent.
3. They are similar because the larger circle has circumference $2.5$ times the smaller circle’s circumference.
4. They are similar because the larger circle encloses the smaller one in the drawing.

Explanation: The skill here is understanding circle similarity in geometry for concentric circles. Circles are similar because a dilation from their common center scales one to the other while preserving the shape. The shared center O allows a straightforward dilation without needing additional translations. Applying a dilation centered at O with scale factor 10/4 directly maps the smaller circle to the larger one, aligning both radius and position. This exact mapping confirms they are similar but not congruent due to the scale factor not being 1, justifying the relationship. A common distractor is choice B, which claims they are congruent, but different radii mean they are not the same size. To solve similar problems, think in terms of transformations centered at key points, not formulas.