Question

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

Answer 1

A) The length of the new radius will be 4 feet

C) The new circumference will be 8 times the original circumference

F) The new area will be 64 times the original area

H) The new area will be 16π square feet

Explanation:

Verify each statement

see the attached figure to better understand the problem

Part A) The length of the new radius will be 4 feet

The statement is true

Because

we know that

The radius of the original circle is r=1/2 ft

The scale factor is 8

so

To find out the length of the new radius multiply the radius of the original circle by the scale factor

(1/2)(8)=4 ft

Part B) The length of the new radius will be 32 feet.

The statement is false

Because, the length of the new radius is 4 ft (see part A)

Part C) The new circumference will be 8 times the original circumference

The statement is true

Because

The original circumference is equal to

`C=\pi ((1)/(2))`

`C=(1)/(2)\pi\ ft`

The new circumference is equal to

`C=\pi (4)`

`C=4\pi\ ft`

Divide
`(4\pi)/((1/2)\pi)=8`

so

the new circumference will be the original circumference multiplied by 8

Part D) The new circumference will be 64 times the original circumference

The statement is false

Because, the new circumference will be 8 times the original circumference (see part C)

Part E) The new area will be 8 times the original area

The statement is false

we know that

If two figures are similar, then the ratio of its areas is equal to the scale factor squared

The scale factor is 8

so

The scale factor squared is 8²=64

therefore

The new area will be 64 times the original area

Part F) The new area will be 64 times the original area

The statement is true

(see Part E)

Part G) The new circumference will be 8π feet

The statement is false

Because, the new circumference will be 4π feet (see part C)

Part H) The new area will be 16π square feet

The statement is true

Because

The area of a circle is

`A=\pi r^(2)`

we have

`r=4\ ft`

substitute

`A=\pi (4)^(2)`

`A=16\pi\ ft^(2)`