Answer :
To determine whether two circles are similar, we need to examine the ratios of their corresponding measures. Specifically, we need to compare the ratios of their radii, diameters, circumferences, or areas. Given the provided choices, we can analyze them based on this criterion.
### Step-by-Step Analysis
1. Option A:
- [tex]\(\frac{r}{r_1}=\frac{4}{3}\)[/tex]
- [tex]\(\frac{C}{C_1}=\frac{16 \pi}{9 \pi}\)[/tex]
For circles to be similar, the ratio of their circumferences should equal the ratio of their radii.
- Simplify the ratio of circumferences: [tex]\(\frac{16 \pi}{9 \pi} = \frac{16}{9}\)[/tex]
- Since [tex]\(\frac{4}{3} \neq \frac{16}{9}\)[/tex], Option A is incorrect.
2. Option B:
- [tex]\(\frac{d}{d_1} = \frac{3}{4}\)[/tex]
- [tex]\(\frac{A}{A_1} = \frac{9 \pi}{16 \pi}\)[/tex]
For circles to be similar, the ratio of their areas should be the square of the ratio of their diameters.
- Simplify the ratio of areas: [tex]\(\frac{9 \pi}{16 \pi} = \frac{9}{16}\)[/tex]
- [tex]\((\frac{3}{4})^2 = \frac{9}{16}\)[/tex], which matches the given area ratio. Hence, Option B is correct.
3. Option C:
- [tex]\(\frac{r}{r_1} = \frac{3}{4}\)[/tex]
- [tex]\(\frac{C}{C_1} = \frac{6 \pi}{8 \pi}\)[/tex]
For circles to be similar, the ratio of their circumferences should equal the ratio of their radii.
- Simplify the ratio of circumferences: [tex]\(\frac{6 \pi}{8 \pi} = \frac{6}{8} = \frac{3}{4}\)[/tex]
- Since [tex]\(\frac{r}{r_1} = \frac{3}{4}\)[/tex] matches the ratio of circumferences, Option C is valid on both criteria. However, let's compare it to Option B:
4. Option D:
- [tex]\(\frac{d}{d_1} = \frac{8}{6} = \frac{4}{3}\)[/tex]
- [tex]\(\frac{A}{A_1} = \frac{16 \pi}{12 \pi} = \frac{4}{3}\)[/tex]
For circles to be similar, the ratio of their areas should be the square of the ratio of their diameters.
- [tex]\((\frac{4}{3})^2 = \frac{16}{9}\)[/tex], which does not match the given area ratio. Hence, Option D is incorrect.
### Conclusion
Option B consistently satisfies the conditions for the circles to be similar through both the diameter and area ratios:
- The ratio of the diameters is [tex]\(\frac{3}{4}\)[/tex], and the ratio of the areas is [tex]\(\frac{9}{16}\)[/tex].
- Since [tex]\((\frac{3}{4})^2 = \frac{9}{16}\)[/tex], the conditions for similarity are met.
Therefore, the correct answer is:
B. [tex]\(\frac{d}{d_1}=\frac{3}{4} ; \frac{A}{A_1}=\frac{9 \pi}{16 \pi}\)[/tex]
### Step-by-Step Analysis
1. Option A:
- [tex]\(\frac{r}{r_1}=\frac{4}{3}\)[/tex]
- [tex]\(\frac{C}{C_1}=\frac{16 \pi}{9 \pi}\)[/tex]
For circles to be similar, the ratio of their circumferences should equal the ratio of their radii.
- Simplify the ratio of circumferences: [tex]\(\frac{16 \pi}{9 \pi} = \frac{16}{9}\)[/tex]
- Since [tex]\(\frac{4}{3} \neq \frac{16}{9}\)[/tex], Option A is incorrect.
2. Option B:
- [tex]\(\frac{d}{d_1} = \frac{3}{4}\)[/tex]
- [tex]\(\frac{A}{A_1} = \frac{9 \pi}{16 \pi}\)[/tex]
For circles to be similar, the ratio of their areas should be the square of the ratio of their diameters.
- Simplify the ratio of areas: [tex]\(\frac{9 \pi}{16 \pi} = \frac{9}{16}\)[/tex]
- [tex]\((\frac{3}{4})^2 = \frac{9}{16}\)[/tex], which matches the given area ratio. Hence, Option B is correct.
3. Option C:
- [tex]\(\frac{r}{r_1} = \frac{3}{4}\)[/tex]
- [tex]\(\frac{C}{C_1} = \frac{6 \pi}{8 \pi}\)[/tex]
For circles to be similar, the ratio of their circumferences should equal the ratio of their radii.
- Simplify the ratio of circumferences: [tex]\(\frac{6 \pi}{8 \pi} = \frac{6}{8} = \frac{3}{4}\)[/tex]
- Since [tex]\(\frac{r}{r_1} = \frac{3}{4}\)[/tex] matches the ratio of circumferences, Option C is valid on both criteria. However, let's compare it to Option B:
4. Option D:
- [tex]\(\frac{d}{d_1} = \frac{8}{6} = \frac{4}{3}\)[/tex]
- [tex]\(\frac{A}{A_1} = \frac{16 \pi}{12 \pi} = \frac{4}{3}\)[/tex]
For circles to be similar, the ratio of their areas should be the square of the ratio of their diameters.
- [tex]\((\frac{4}{3})^2 = \frac{16}{9}\)[/tex], which does not match the given area ratio. Hence, Option D is incorrect.
### Conclusion
Option B consistently satisfies the conditions for the circles to be similar through both the diameter and area ratios:
- The ratio of the diameters is [tex]\(\frac{3}{4}\)[/tex], and the ratio of the areas is [tex]\(\frac{9}{16}\)[/tex].
- Since [tex]\((\frac{3}{4})^2 = \frac{9}{16}\)[/tex], the conditions for similarity are met.
Therefore, the correct answer is:
B. [tex]\(\frac{d}{d_1}=\frac{3}{4} ; \frac{A}{A_1}=\frac{9 \pi}{16 \pi}\)[/tex]
