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

Explanation:

It is given that the radius of circle before dilation is
`(1)/(2)ft` and the scale factor is 8.

The circumference of original circle is,

`S_1=2\pi r`

`S_1=2\pi * (1)/(2)=\pi`

The area of original circle is,

`A_1=\pi r^2`

`A_1=\pi ((1)/(2))^2`

`A_1=(\pi)/(4)`

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

`r=8* (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 * 4=8\pi`

`(S_2)/(S_1)=(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`

`(A_2)/(A_1)=(16\pi)/((\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.