Respuesta :
Answer: The area of the dilated circle will be one-forth of the area of the original circle.
Step-by-step explanation: We are given to find the change in area of a circle if it is dilated by a scale factor of half.
Let r represents the RADIUS and A represents the AREA of the circle before dilation.
And, r' represents the radius and A' represents the AREA of the circle after dilation.
So, we get
[tex]A=\pi r^2~~~~~~\textup{and}~~~~~~A'=\pi r'^2.[/tex]
In case of a circle, scale factor is the ratio of the radius of the dilated circle to the radius of the original circle.
That is,
[tex]\dfrac{1}{2}=\dfrac{r'}{r}\\\\\\\Rightarrow r'=\dfrac{r}{2}.[/tex]
Therefore, the area of the dilated circle will be
[tex]A'=\pi r'^2=\pi \times\left(\dfrac{r}{2}\right)^2=\dfrac{\pi r^2}{4}=\dfrac{1}{4}\times A.[/tex]
Thus, the area of the dilated circle will be one-forth of the area of the original circle.
