Question

two circles are shown in the diagram.
c₁ (small circle) with ( d_1 = 1 ), c₂ (large circle) with ( d_2 = 2r ).
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 )?

Explanation:

Step1: Substitute given values into proportion

We know \( C_1 = \pi \), \( d_1 = 1 \), and \( d_2 = 2r \). Substitute these into the proportion \(\frac{C_1}{d_1}=\frac{C_2}{d_2}\), 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 gives us \(1\times C_2=\pi\times2r\), which simplifies to \(C_2 = 2\pi r\).

Answer:

To solve for \( C_2 \), substitute \( C_1=\pi \), \( d_1 = 1 \), and \( d_2 = 2r \) into \(\frac{C_1}{d_1}=\frac{C_2}{d_2}\) to get \(\frac{\pi}{1}=\frac{C_2}{2r}\), then cross - multiply to find \( C_2 = 2\pi r \). The circumference of a circle with radius \( r \) is \( C_2=\boxed{2\pi r}\).