Question

a transformation was applied to a circle. the image of the transformation has the same center as the circle. which statement is true about the transformation? the transformation is rigid because the pre - image and image are congruent. the transformation could be a reflection because the circles do not overlap. the transformation is not a translation because the circles share a center. the transformation cannot be isometric because the circles are not the same size.

Explanation:

1. Analyze Option 1: A rigid transformation (like translation, reflection, rotation) preserves congruence (same size and shape). The circles here have different sizes (one dashed, one solid, different radii), so they aren’t congruent. Thus, this statement is false.
2. Analyze Option 2: A reflection would map a circle to a congruent circle (same size) and the center’s reflection would depend on the line of reflection. Since the circles have different sizes, reflection is not the case, and the “do not overlap” reason is irrelevant. So this is false.
3. Analyze Option 3: A translation moves a figure without rotating or resizing, and the center would move (since translation shifts all points by the same vector). Here, the centers are the same, so it can’t be a translation. This statement is true.
4. Analyze Option 4: Isometric transformations (rigid motions) preserve distance (and thus size). But a dilation (non - isometric) changes size. However, the statement says “cannot be isometric because circles are not same size” – but an isometric transformation would keep them same size. But the key is that the transformation here is likely a dilation (changes size), but the option’s reasoning is about “cannot be isometric” which is correct in a way, but the other option (Option 3) is more clearly correct. Wait, no – let's re - check. The first circle (solid) and the image (dashed) have different radii. A translation is a rigid motion (isometric) that moves the figure. Since the center is the same, translation (which moves all points, including the center) is impossible. So Option 3 is correct. The other options: Option 1 is wrong (not congruent), Option 2 is wrong (reflection would keep size same), Option 4: Isometric transformations preserve size, so if the circles are different sizes, the transformation isn’t isometric, but the reason in Option 4 is “because the circles are not the same size” – but the main point for Option 3 is about translation. Since translation requires moving the center (as it’s a shift), and here centers are same, so translation is impossible. So Option 3 is correct.

Answer:

The transformation is not a translation because the circles share a center.