Question

circle f and diameter \\( \overline { e g } \\) are shown on the graph below. circle f is dilated by a scale factor of \\( \frac { 1 } { 4 } \\) centered at the origin to create circle \\( f ^ { \prime } \\).

what is the circumference of circle \\( f ^ { \prime } \\)?

write your answer as a whole number or as a decimal rounded to the nearest tenth.

units

Explanation:

Step1: Find the diameter of circle \( F \)

From the graph, the coordinates of \( E(-8,-3) \) and \( G(4,-3) \).
Using the distance formula for points on a horizontal line (\( y - \)coordinate is the same), \( d=\vert x_2 - x_1\vert\).
So, diameter of circle \( F\), \(d = 4-(-8)=12\) units.

Step2: Find the diameter of circle \( F'\)

Since circle \( F \) is dilated by a scale factor of \( \frac{1}{4} \), the diameter of circle \( F'\) is \(d'=12\times\frac{1}{4} = 3\) units.

Step3: Calculate the circumference of circle \( F'\)

The formula for the circumference of a circle is \(C=\pi d\).
Substituting \(d = 3\) into the formula, \(C=\pi\times3\approx3.14\times 3 = 9.4\) units.

Answer:

\(9.4\) units