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Geometry Question of the Day

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Friday, October 23, 2026

Two circles are shown: $\odot O$ has center $O$ and radius $4$ , and $\odot P$ has center $P$ and radius $10$ . Which transformation maps $\odot O$ to a circle similar to $\odot P$ ?

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Two circles are shown: $\odot O$ has center $O$ and radius $4$ , and $\odot P$ has center $P$ and radius $10$ . Which transformation maps $\odot O$ to a circle similar to $\odot P$ ?

Translate $\odot O$ so that $O$ moves to $P$ , then dilate by scale factor $\tfrac{5}{2}$ . (correct answer)
Rotate $\odot O$ about $O$ by $90^\circ$ to match the size of $\odot P$ .
Reflect $\odot O$ across line $\overline{OP}$ to make its radius equal to $10$ .
Use $A=\pi r^2$ to show the area ratio is $\left(\tfrac{5}{2}\right)^2$ , so no transformation is needed.

Explanation: The skill here is understanding circle similarity. Circles are similar because one can be mapped to another via a dilation that scales all distances by a constant factor, preserving shape. In this problem, the circles have centers O and P with radii 4 and 10, respectively. Applying a translation to move O to P, followed by a dilation centered at P with scale factor 5/2, maps ⊙O directly to ⊙P. This justifies similarity because the composition of translation (an isometry) and dilation is a similarity transformation that maps one circle exactly onto the other. A distractor like choice B suggests rotation changes size, but rotations preserve distances and do not alter radii. To approach similar problems, think in terms of transformations like dilations rather than relying on formulas.