Answer :
Let's analyze the argument given and identify the step that contains an error and how it can be corrected.
1. Step 1:
- Translate circle B to the right 9 units and up 7 units to form concentric circles.
- This step is correct. Translating circle B by moving it to the right 9 units and up 7 units correctly moves the center from [tex]$(-4, -5)$[/tex] to [tex]$(5, 2)$[/tex]. Note that this places the center of circle B at [tex]$(5, 2)$[/tex], not at [tex]$(3, 4)$[/tex], which would align it with circle A. Nonetheless, this step achieves the goal of translating circle B.
2. Step 2:
- Dilate circle B to be congruent to circle A using a scale factor of \( z = \frac{3}{7} - \frac{8}{3} \).
- This step is incorrect. The dilation factor given, \( z = \frac{3}{7} - \frac{8}{3} \), is not correct. To make circle B congruent to circle A, we need to adjust its radius to match that of circle A. This can be done by using the ratio of the radii of circle A and circle B.
- Circle A has a radius of 2, and circle B has a radius of 3. Therefore, the correct scale factor should be the ratio of circle A's radius to circle B's radius:
[tex]\[ \text{Correct scale factor} = \frac{\text{Radius of Circle A}}{\text{Radius of Circle B}} = \frac{2}{3}. \][/tex]
- So, circle B should be dilated using a scale factor of \( \frac{2}{3} \).
3. Step 3:
- When an object is dilated, the dilated object is similar to the preimage, thus the two circles are similar.
- This step is also correct in that dilation does preserve similarity (not necessarily congruence). Since a dilation by any positive scale factor results in a figure similar to the original, this reasoning stands valid once the correct scale factor is used.
Conclusion:
- Step 2 is the incorrect step.
- It should be corrected by dilating circle B using the scale factor \( \frac{2}{3} \).
So, the correct answer is:
Step 2, circle B should have been dilated using a scale factor of [tex]\( \frac{2}{3} \)[/tex].
1. Step 1:
- Translate circle B to the right 9 units and up 7 units to form concentric circles.
- This step is correct. Translating circle B by moving it to the right 9 units and up 7 units correctly moves the center from [tex]$(-4, -5)$[/tex] to [tex]$(5, 2)$[/tex]. Note that this places the center of circle B at [tex]$(5, 2)$[/tex], not at [tex]$(3, 4)$[/tex], which would align it with circle A. Nonetheless, this step achieves the goal of translating circle B.
2. Step 2:
- Dilate circle B to be congruent to circle A using a scale factor of \( z = \frac{3}{7} - \frac{8}{3} \).
- This step is incorrect. The dilation factor given, \( z = \frac{3}{7} - \frac{8}{3} \), is not correct. To make circle B congruent to circle A, we need to adjust its radius to match that of circle A. This can be done by using the ratio of the radii of circle A and circle B.
- Circle A has a radius of 2, and circle B has a radius of 3. Therefore, the correct scale factor should be the ratio of circle A's radius to circle B's radius:
[tex]\[ \text{Correct scale factor} = \frac{\text{Radius of Circle A}}{\text{Radius of Circle B}} = \frac{2}{3}. \][/tex]
- So, circle B should be dilated using a scale factor of \( \frac{2}{3} \).
3. Step 3:
- When an object is dilated, the dilated object is similar to the preimage, thus the two circles are similar.
- This step is also correct in that dilation does preserve similarity (not necessarily congruence). Since a dilation by any positive scale factor results in a figure similar to the original, this reasoning stands valid once the correct scale factor is used.
Conclusion:
- Step 2 is the incorrect step.
- It should be corrected by dilating circle B using the scale factor \( \frac{2}{3} \).
So, the correct answer is:
Step 2, circle B should have been dilated using a scale factor of [tex]\( \frac{2}{3} \)[/tex].
