Question

since all circles are similar, a proportion can be set up using the circumference and diameter of each circle substitute the values $d_1 = 1$, $c_1 = pi$, and $d_2 = 2r$ into the proportion.\\(\frac{c_1}{d_1} = \frac{c_2}{d_2}\\)\
which shows how to correctly solve for $c_2$, the circumference of any circle with radius $r$?\
\\(\circ\\) because \\(\frac{pi}{1} = \frac{c_2}{2r}\\), $c_2 = 2rpi$\
\\(\circ\\) because \\(\frac{1}{pi} = \frac{c_2}{2r}\\), $c_2 = \frac{2r}{pi}$\
\\(\circ\\) because \\(\frac{pi}{2r} = \frac{c_2}{1}\\), $c_2 = \frac{pi}{2r}$\
\\(\circ\\) because \\(\frac{pi}{1} = \frac{c_2}{4r}\\), $c_2 = 4rpi$

Explanation:

Step1: Substitute values into proportion

Given \( \frac{C_1}{d_1} = \frac{C_2}{d_2} \), substitute \( C_1 = \pi \), \( d_1 = 1 \), \( d_2 = 2r \). So we get \( \frac{\pi}{1} = \frac{C_2}{2r} \).

Step2: Solve for \( C_2 \)

Cross - multiply the proportion \( \frac{\pi}{1}=\frac{C_2}{2r} \). Cross - multiplying means \( \pi\times2r = C_2\times1 \), which simplifies to \( C_2 = 2r\pi \).

Answer:

Because \( \frac{\pi}{1}=\frac{C_2}{2r}, C_2 = 2r\pi \)