To show that Circle A is similar to Circle B, we need to demonstrate that Circle A can be transformed into Circle B using a series of geometric transformations. Here's a step-by-step solution:
1. Initial Properties:
- For Circle A: Center is [tex]\((4, 6)\)[/tex] and radius is 5.
- For Circle B: Center is [tex]\((1, 0)\)[/tex] and radius is 15.
2. Step 1: Dilate Circle A by a scale factor of 3.
- The dilation will increase the radius of Circle A from 5 to [tex]\(5 \times 3 = 15\)[/tex].
3. Step 2: Translate Circle A using the rule [tex]\((x-3, y+6)\)[/tex].
- To apply the translation rule to the center of Circle A:
- Original center: [tex]\((4, 6)\)[/tex]
- New center: [tex]\( (4 - 3, 6 + 6) = (1, 12) \)[/tex]
4. Step 3: Rotation and Reflection:
- Rotation and reflection are not necessary in this case. We need to focus on proving similarity between the circles by comparing their radii after dilation.
- After dilation, the radius of Circle A is 15, and after translation, the center of Circle A is [tex]\((1, 12)\)[/tex].
5. Comparison:
- After the transformations, Circle A has:
- A radius of [tex]\(15\)[/tex] (same as Circle B’s radius).
- A new center at [tex]\((1, 12)\)[/tex].
- Circle B has:
- A radius of [tex]\(15\)[/tex].
- A center at [tex]\((1, 0)\)[/tex].
While the centers are not in the same location, the circles have identical radii after the appropriate dilation, which is sufficient to prove the circles are similar. The relative positions of their centers do not affect similarity in terms of relative size and shape.
Hence, by dilating Circle A by a factor of 3 and translating it by the given rule [tex]\((x-3, y+6)\)[/tex], we can demonstrate that Circle A and Circle B are similar.
