Question

which sequence of transformations could best be used to verify that the two figures are similar to each other?
c. dilate the shaded figure with scale factor of \\(\frac{1}{2}\\) centered at the origin, and then rotate clockwise about the origin.
b. dilate the shaded figure with scale factor of \\(\frac{1}{2}\\) centered at the origin, and then reflect across the y - axis.
d. dilate the shaded figure with scale factor of \\(\frac{1}{3}\\) centered at the origin, and then rotate counterclockwise about the origin.
dilate the shaded figure with scale factor of \\(\frac{1}{3}\\) centered at the origin, and then reflect across the x - axis.

Explanation:

Step1: Analyze the size relationship

The shaded figure is larger, and the unshaded one is smaller. To make them similar, we need to reduce the shaded figure. The scale factor should be such that the size matches. Observing the options, a scale factor of \( \frac{1}{3} \) or \( \frac{1}{2} \)? Wait, let's check the orientation. The unshaded figure is rotated? Wait, no, let's see the position. Wait, the shaded figure and the unshaded: when we dilate the shaded figure with scale factor \( \frac{1}{3} \) (since the unshaded is about 1/3 the size of the shaded), then rotate? Wait, no, option D: Dilate with scale factor \( \frac{1}{3} \), centered at origin, then reflect across x - axis? Wait, no, maybe I made a mistake. Wait, let's re - evaluate. Wait, the shaded figure is larger, the unshaded is smaller. Let's check the scale factor. If the shaded figure is dilated by \( \frac{1}{3} \), it would match the size of the unshaded. Then, about the rotation or reflection. Wait, the unshaded figure's orientation: if we dilate the shaded figure with scale factor \( \frac{1}{3} \) centered at the origin, then rotate counter - clockwise? Wait, no, option D: Dilate with scale factor \( \frac{1}{3} \), centered at origin, then rotate counter - clockwise? Wait, no, the correct sequence: the shaded figure is larger, the unshaded is smaller. So we need to dilate the shaded figure by a scale factor that reduces its size to match the unshaded. Let's assume the shaded figure's dimensions are 3 times the unshaded. So scale factor \( \frac{1}{3} \). Then, the rotation: if we rotate the dilated shaded figure counter - clockwise about the origin, it would match the unshaded's orientation. Wait, option D: Dilate the shaded figure with scale factor of \( \frac{1}{3} \) centered at the origin, and then rotate counterclockwise about the origin. Wait, but let's check the options again. Wait, maybe the correct answer is D? Wait, no, let's re - check. Wait, the unshaded figure is below the x - axis? No, the shaded is above, unshaded is near x = 2, y negative? Wait, maybe the reflection? No, let's think about similar figures: similarity transformations include dilation, rotation, reflection, translation. To verify similarity, we can perform a dilation (to make them same size) and then a rotation/reflection/translation to map one to the other. The shaded figure is larger, so we need to dilate it by a scale factor less than 1. The unshaded figure seems to be about 1/3 the size of the shaded, so scale factor \( \frac{1}{3} \). Then, the rotation: if we rotate the dilated shaded figure counter - clockwise about the origin, it would match the unshaded's orientation. So option D: Dilate the shaded figure with scale factor of \( \frac{1}{3} \) centered at the origin, and then rotate counterclockwise about the origin. Wait, but let's check the options again. Wait, the options:

A: Not shown, B: scale factor \( \frac{1}{2} \), reflect across y - axis, C: scale factor \( \frac{1}{2} \), rotate clockwise, D: scale factor \( \frac{1}{3} \), rotate counterclockwise, and the last option: scale factor \( \frac{1}{3} \), reflect across x - axis.

Wait, maybe the correct answer is D. Because the scale factor of \( \frac{1}{3} \) is more appropriate (since the unshaded figure is much smaller, about 1/3 the size of the shaded), and then rotating counterclockwise about the origin would map the dilated shaded figure to the unshaded one.

Step2: Confirm the transformation sequence

To verify similarity, we need to show that one figure can be obtained from the other by a similarity transformation (dilation + rigid motion: rotation, reflection, translation). The shaded figure is larger, so we dilate it with a scale factor to reduce its size. The unshaded figure's size relative to the shaded suggests a scale factor of \( \frac{1}{3} \). Then, a rotation (counterclockwise about the origin) would align the dilated figure with the unshaded one. So the sequence in option D (Dilate the shaded figure with scale factor of \( \frac{1}{3} \) centered at the origin, and then rotate counterclockwise about the origin) is the correct one.

Answer:

D. Dilate the shaded figure with scale factor of \( \frac{1}{3} \) centered at the origin, and then rotate counterclockwise about the origin.