To show that Circle 1 and Circle 2 are similar, we need to perform two transformations: a translation and a dilation.
1. Translation:
We need to translate Circle 1 so that its center matches the center of Circle 2. The translation vector can be found by subtracting the coordinates of the center of Circle 1 from the coordinates of the center of Circle 2:
- Translation in the x-direction: [tex]\(1 - 5 = -4\)[/tex]
- Translation in the y-direction: [tex]\(-2 - 8 = -10\)[/tex]
Therefore, the translation rule is [tex]\((x - 4, y - 10)\)[/tex].
2. Dilation:
After translating Circle 1, we need to scale it to match the size of Circle 2. The scale factor for dilation can be determined by the ratio of the radii of the two circles. Circle 1 has a radius of 8 cm, and Circle 2 has a radius of 4 cm. The scale factor (k) is:
[tex]\[ k = \frac{\text{Radius of Circle 2}}{\text{Radius of Circle 1}} = \frac{4}{8} = 0.5 \][/tex]
Putting it all together:
The circles are similar because you can translate Circle 1 using the transformation rule [tex]\((x - 4, y - 10)\)[/tex] and then dilate it using a scale factor of [tex]\(0.5\)[/tex].
