Two circles are concentric. Two arcs are formed by the same central angle, with arc lengths of 54 an

Two circles are concentric. Two arcs are formed by the same central angle, with arc lengths of 54 and 81 respectively. Which of the following are possible radii for the circles? so there are two possible answers.
- 14 and 41
- 23 and 32
- 32 and 64
- 40 and 60
- 12 and 18
I believe 12 and 18 are one.
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Vikki Chapman
Consider ${r}_{1}=$ radius of inner circle
${r}_{2}=$ radius of outer circle
${S}_{1}=$ arc length of inner circle
${S}_{2}=$ arc length of outer circle.
$\theta =$ central angle for both arcs .
relation s,r and $\theta$ is
$S=r\theta$

Let $\frac{{r}_{1}}{{r}_{2}}=\frac{\frac{{S}_{1}}{\theta }}{\frac{{S}_{2}}{\theta }}=\frac{{S}_{1}}{{S}_{2}}=\frac{54}{81}=\frac{27×2}{27×3}=\frac{2}{3}$
Since ${S}_{1}=54$
${S}_{2}=81$
We have radius ratio $\frac{2}{3}$
We check whether which options satisfy ratio
1) ${r}_{1}=14,{r}_{2}=41⇒\frac{{r}_{1}}{{r}_{2}}=\frac{14}{41}\ne \frac{2}{3}$
2) ${r}_{1}=23,{r}_{2}=32⇒\frac{{r}_{1}}{{r}_{2}}=\frac{23}{32}\ne \frac{2}{3}$
3) ${r}_{1}=40,{r}_{2}=64⇒\frac{{r}_{1}}{{r}_{2}}=\frac{32}{64}=\frac{16×2}{16×4}=\frac{2×1}{2×2}=\frac{1}{2}\ne \frac{2}{3}$
4) ${r}_{1}=40,{r}_{2}=60⇒\frac{{r}_{1}}{{r}_{2}}=\frac{40}{60}=\frac{4}{6}=\frac{2×2}{2×3}=\frac{2}{3}$
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