Question

1. $\frac{c = 2pi r}{c=2pi r}$ and $\frac{c}{c}=\frac{r}{r}$ by the division property of equality.
2. $d = 2r$ and $d = 2r$ by the definition of diameter.
3. $\frac{d = 2r}{d=2r}$ and $\frac{d}{d}=\frac{r}{r}$ by the division property of equality.
4. $\frac{c}{c}=\frac{d}{d}$ and $\frac{r}{r}=\frac{r}{r}$ by the substitution property.
5. circle $o$ is similar to circle $x$ because all the linear dimensions are in the same proportion.

a. $c=pi r$ and $c=pi r$ by the definition of area of a circle.
b. $c = 2r$ and $c=2r$ by the definition of circumference.
c. $c = 2pi r$ and $c=2pi r$ by the definition of circumference.
d. $c=pi r^{2}$ and $c=pi(r)^{2}$ by the definition of area of a circle.

Explanation:

Step1: Recall circle - circumference formula

The formula for the circumference of a circle is $C = 2\pi r$, where $C$ is the circumference and $r$ is the radius. For two circles with radii $r$ and $r'$, their circumferences are $C = 2\pi r$ and $C'=2\pi r'$ respectively.

Step2: Analyze each option

• Option A: The formula $C=\pi r$ is incorrect for the circumference of a circle. The correct formula for the area of a circle is $A = \pi r^{2}$.
• Option B: The formula $C = 2r$ is incorrect for the circumference of a circle. The correct formula for the circumference is $C=2\pi r$.
• Option C: $C = 2\pi r$ and $C'=2\pi r'$ are the correct formulas for the circumferences of two circles with radii $r$ and $r'$ respectively, based on the definition of the circumference of a circle.
• Option D: $C=\pi r^{2}$ is the formula for the area of a circle, not the circumference.

Answer:

C. $C = 2\pi r$ and $C'=2\pi r'$ by the definition of circumference.