All Circles Are Similar
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Geometry › All Circles Are Similar
A circle $\odot C$ is dilated about its center $C$ by a scale factor $k>0$ to produce a new circle $\odot C'$. The centers are marked. Which reasoning uses similarity correctly?
The dilation changes the shape of the circle, so the image is not similar to the original.
The circles are similar because their circumferences both involve $2\pi$.
The dilation preserves shape, so the image circle is similar to the original circle.
The circles are similar only if $k=1$, because then they are the same size.
Explanation
The skill being assessed is understanding that all circles are similar. Similarity means that shapes are the same but possibly different sizes, preserved under transformations like dilation. A dilation can map one circle to another by scaling the radius while keeping the circular shape intact. Here, the dilation is about center C, producing ⊙C' with a possibly different radius, but the center remains marked at C for both. The correct reasoning in choice B justifies similarity by emphasizing that dilation preserves the circular shape. A common misconception, as in choice C, is thinking similarity requires the same size (k=1), but similarity allows for scaling. To understand circle similarity, think in terms of transformations like dilation and translation, rather than formulas for circumference or area.
Two circles $\odot G$ and $\odot H$ are drawn with different radii. Their centers are marked at $G$ and $H$. Which property guarantees circle similarity?
Circles are similar because a dilation preserves shape while scaling distances.
Circles are similar only when they overlap in the diagram.
Circles are similar because each has a diameter that is twice its radius.
Circles are similar because their radii have a constant difference.
Explanation
The skill being assessed is understanding that all circles are similar. Similarity means that shapes are the same but possibly different sizes, preserved under transformations like dilation. A dilation can map one circle to another by scaling the radius while keeping the circular shape intact. Here, centers G and H are marked, and the circles have different radii, demonstrating that position and size differ but shape remains. The correct reasoning in choice B justifies similarity by noting dilation preserves shape during scaling. A common misconception, as in choice A, is believing constant radius differences imply similarity, but differences are irrelevant to shape. To understand circle similarity, think in terms of transformations like dilation and translation, rather than formulas for circumference or area.
Two circles are shown: $\odot O$ with center $O$ and $\odot P$ with center $P$, with different radii. Which description relies on dilation to justify that the circles are similar?
Use the fact that both circles contain infinitely many points to conclude they are similar.
Rotate $\odot O$ about $O$ until it overlaps $\odot P$ and the radii become equal.
Dilate $\odot O$ about $O$ to change its radius, then translate to align the centers.
Translate $\odot O$ so that $O$ lands on $P$, and the circles will match exactly.
Explanation
The skill being assessed is understanding that all circles are similar. Similarity means that shapes are the same but possibly different sizes, preserved under transformations like dilation. A dilation can map one circle to another by scaling the radius while keeping the circular shape intact. Here, centers O and P are marked, and the circles have different radii, necessitating scaling and repositioning. The correct reasoning in choice B justifies similarity by using dilation to adjust radius, followed by translation to align centers. A common misconception, as in choice A, is thinking translation alone suffices, but it does not change size. To understand circle similarity, think in terms of transformations like dilation and translation, rather than formulas for circumference or area.