To determine the new length of the radius of the circle after it has been dilated by a scale factor of [tex]\(\frac{1}{4}\)[/tex], we follow these steps:
1. Identify the initial radius of the circle: Suppose the initial radius of the circle is 20 cm.
2. Understand the meaning of the scale factor: A scale factor of [tex]\(\frac{1}{4}\)[/tex] means that each dimension of the circle is reduced to [tex]\(\frac{1}{4}\)[/tex] of its original length.
3. Calculate the new radius: To find the new radius after the dilation, multiply the initial radius by the scale factor:
[tex]\[
\text{New Radius} = \text{Initial Radius} \times \text{Scale Factor}
\][/tex]
Plugging in the known values:
[tex]\[
\text{New Radius} = 20 \, \text{cm} \times \frac{1}{4}
\][/tex]
Simplifying the multiplication:
[tex]\[
\text{New Radius} = 5 \, \text{cm}
\][/tex]
Therefore, the length of the new radius after the circle is dilated by a scale factor of [tex]\(\frac{1}{4}\)[/tex] is [tex]\(\boxed{5 \text{ cm}}\)[/tex].
