Question

use the diagram for problems 6 - 7.

1. if ( odot t ) is the pre - image and ( odot f ) is the image, how would the scale factor be different?
2. find the indicated ratios.

a. ( \frac{\text{radius } odot f}{\text{radius } odot t}= )
b. ( \frac{\text{circumference } odot f}{\text{circumference } odot t}= )
c. ( \frac{moverset{\frown}{gh}}{moverset{\frown}{rx}}= )
d. ( \frac{\text{area } odot f}{\text{area } odot t}= )
e. ( \frac{\text{area sector } gfh}{\text{area sector } rtx}= )

Explanation:

Step1: Find the ratio of radii

The radius of $\odot F$ is $2$ cm and the radius of $\odot T$ is $6$ cm.
$$\frac{\text{radius }\odot F}{\text{radius }\odot T}=\frac{2}{6}=\frac{1}{3}$$

Step2: Find the ratio of circumferences

The formula for the circumference of a circle is $C = 2\pi r$.
$$\frac{\text{circumference }\odot F}{\text{circumference }\odot T}=\frac{2\pi\times2}{2\pi\times6}=\frac{2}{6}=\frac{1}{3}$$

Step3: Find the ratio of arc measures

The measure of an arc is related to the central angle. Since the central angles for $\overset{\frown}{GH}$ and $\overset{\frown}{RX}$ are both $135^{\circ}$.
$$\frac{m\overset{\frown}{GH}}{m\overset{\frown}{RX}}=\frac{135^{\circ}}{135^{\circ}} = 1$$

Step4: Find the ratio of areas

The formula for the area of a circle is $A=\pi r^{2}$.
$$\frac{\text{area }\odot F}{\text{area }\odot T}=\frac{\pi\times2^{2}}{\pi\times6^{2}}=\frac{4}{36}=\frac{1}{9}$$

Step5: Find the ratio of sector areas

The formula for the area of a sector is $A=\frac{\theta}{360}\pi r^{2}$ (where $\theta$ is the central angle). Since $\theta = 135^{\circ}$ for both sectors.
$$\frac{\text{area sector }GFH}{\text{area sector }RTX}=\frac{\frac{135}{360}\pi\times2^{2}}{\frac{135}{360}\pi\times6^{2}}=\frac{4}{36}=\frac{1}{9}$$

Answer:

a. $\frac{1}{3}$
b. $\frac{1}{3}$
c. $1$
d. $\frac{1}{9}$
e. $\frac{1}{9}$