Question

A circle with a radius of 1/2 ft is dilated by a scale factor of 8. Which statements about the new circle are true? Check all that apply.
A.The length of the new radius will be 4 feet.
B. The length of the new radius will be 32 feet.
C.The new circumference will be 8 times the original circumference.
D.The new circumference will be 64 times the original circumference.
E.The new area will be 8 times the original area.
F.The new area will be 64 times the original area.
G.The new circumference will 8PI be
H.The new area will be 16PI square feet.

Answer 1

The circle is being dilated by a factor of 8 so the radius will be 8 times the original radius (4 ft).

The circumference of a circle = $2\pi r$, so if r is replaced by $8r$ we get $C = 2\pi * 8r$ or just $C = 8*(2\pi r)$.

Basically the circumference will be 8 times the original circumference.

If you plug in the original radius, into this new formula you get $C_{new} = 8*(2\pi * \frac{1}{2}) = 8\pi \: ft$

The area of a circle = $\pi r^2$, so if r is replaced by $8r$ we get $A = \pi*(8r)^2$ or just $A = 64\pi r^2$

Basically the area will be 64 times the original area.

If you plug in the original radius into this new formula you get $A = 64\pi * (\frac{1}{2})^2 = 16\pi \: ft^2$

The answers are A, C, F, G, and H

Answer 2

Answer:

The statements A,C,F,G and H are true.

Step-by-step explanation:

It is given that the radius of circle before dilation is $\frac{1}{2}ft$ and the scale factor is 8.

The circumference of original circle is,

$S_1=2\pi r$

$S_1=2\pi \times \frac{1}{2}=\pi$

The area of original circle is,

$A_1=\pi r^2$

$A_1=\pi (\frac{1}{2})^2$

$A_1=\frac{\pi}{4}$

The dilation by scale factor 8 means the radius of new circle is 8 times of the original circle.

$r=8\times \frac{1}{2}$

Therefore the radius of new circle is 4 ft and the statement A is true.

The circumference of original circle is,

$S_2=2\pi r$

$S_2=2\pi \times 4=8\pi$

$\frac{S_2}{S_1}=\frac{8\pi}{\pi} =8$

The new circumference will be 8 times the original circumference. The statement C is true.

The area of original circle is,

$A_2=\pi r^2$

$A_2=\pi (4)^2$

$A_2=16\pi$

$\frac{A_2}{A_1}=\frac{16\pi}{\frac{\pi}{4}}=64$

The new area will be 64 times the original area. Therefore statement F is true.

The new circumference will $8\pi$,The new area will be  $16\pi$ square feet.