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Since all circles are similar, a proportion can be set up using the circumference and diameter of each circle. Substitute the values [tex]d_1=1[/tex], [tex]C_1=\pi[/tex], and [tex]d_2=2r[/tex] into the proportion.

[tex]
\frac{C_1}{d_1}=\frac{C_2}{d_2}
[/tex]

Which shows how to correctly solve for [tex]C_2[/tex], the circumference of any circle with radius [tex]r[/tex]?

A. Because [tex]\frac{\pi}{1}=\frac{C_2}{2r}[/tex], [tex]C_2=2r\pi[/tex]

B. Because [tex]\frac{1}{\pi}=\frac{C_2}{2r}[/tex], [tex]C_2=\frac{2r}{\pi}[/tex]

C. Because [tex]\frac{\pi}{2r}=\frac{C_2}{1}[/tex], [tex]C_2=\frac{\pi}{2r}[/tex]

D. Because [tex]\frac{\pi}{1}=\frac{C_2}{4r}[/tex], [tex]C_2=4r\pi[/tex]



Answer :

GinnyAnswer
To solve for [tex]\( C_2 \)[/tex], the circumference of any circle with radius [tex]\( r \)[/tex], we should follow these steps:

1. Understand the given proportion: The proportion [tex]\( \frac{C_1}{d_1} = \frac{C_2}{d_2} \)[/tex] relates the circumference and diameter of two circles. Here, the diameter of the first circle [tex]\( d_1 = 1 \)[/tex] and its circumference [tex]\( C_1 = \pi \)[/tex]. The second circle has a diameter [tex]\( d_2 = 2r \)[/tex] and we need to find its circumference [tex]\( C_2 \)[/tex].

2. Substitute the given values:
[tex]\[ \frac{C_1}{d_1} = \frac{C_2}{d_2} \][/tex]
Substituting [tex]\( C_1 = \pi \)[/tex], [tex]\( d_1 = 1 \)[/tex], and [tex]\( d_2 = 2r \)[/tex]:
[tex]\[ \frac{\pi}{1} = \frac{C_2}{2r} \][/tex]

3. Solve for [tex]\( C_2 \)[/tex]:
[tex]\[ \frac{\pi}{1} = \frac{C_2}{2r} \][/tex]
Multiply both sides by [tex]\( 2r \)[/tex] to isolate [tex]\( C_2 \)[/tex]:
[tex]\[ \pi \cdot 2r = C_2 \][/tex]
Simplify:
[tex]\[ C_2 = 2r\pi \][/tex]

Therefore, the correct solution is:
[tex]\[ \boxed{\text{Because } \frac{\pi}{1} = \frac{C_2}{2r}, \text{ then } C_2 = 2r\pi.} \][/tex]

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