To answer the question of how the circumference of a circle changes when it is dilated by a scale factor of 4, let's carefully analyze the properties of dilations and circles.
1. Understanding dilation: When a geometric shape is dilated by a scale factor [tex]\( k \)[/tex], every linear dimension of the shape (such as lengths, widths, and perimeters) is multiplied by that scale factor [tex]\( k \)[/tex].
2. Circle and its circumference: The circumference [tex]\( C \)[/tex] of a circle is directly proportional to its radius [tex]\( r \)[/tex], given by the formula:
[tex]\[
C = 2 \pi r
\][/tex]
Here, [tex]\( \pi \)[/tex] is a constant and [tex]\( r \)[/tex] is the radius.
3. Effect of dilation on the radius: When the circle is dilated by a scale factor of 4, the new radius [tex]\( r' \)[/tex] becomes:
[tex]\[
r' = 4r
\][/tex]
4. Effect of dilation on the circumference: Since the circumference is directly proportional to the radius, if the radius is multiplied by 4, the circumference will also be multiplied by 4:
[tex]\[
C' = 2 \pi r' = 2 \pi (4r) = 4 (2 \pi r) = 4C
\][/tex]
Therefore, the circumference after dilation will be 4 times greater than the original circumference.
To summarize:
- The circumference will be 4 times greater than the original.
So the correct answer is:
- The circumference will be 4 times greater than the original.
