Question

Two concentric circles are shown. The length of the arc MN is 37 cm and the length of the arc RE is 6 cm.Which statements are correct? Select all that apply.The radius of the larger circle is 18 cm.The radius of the smaller circle is 18 cm.The length of the segment NE is 9 cm.The length of the segment MR is 18 cmThe circumference of the larger circle is 72 cm.The circumference of the larger circle is 367 cmMacBook

Answer 1

- We are asked questions regarding the radii and circumference of the concentric circles.

- Thus, we should try to find the values of the radii and circumferences of the circles.

- The relevant formulas we will use are:

`\begin{gathered} Circumference\text{ of an Arc:} \\ (\theta)/(360)*2\pi r \\ \\ \text{Circumference of a Circle} \\ 2\pi r \end{gathered}`

The Radii:

- We can get the radii using the formula for the circumference of an arc given above.

`\begin{gathered} \text{Smaller circle:} \\ \theta=30 \\ (\theta)/(360)*2\pi r=3\pi \\ \\ (30)/(360)*2\pi r=3\pi \\ \\ (\pi r)/(6)=3\pi \\ \\ r=6*3=18 \\ \text{The radius of the smaller circle is 18}cm \\ \\ \\ \text{Larger Circle:} \\ \theta=30 \\ (30)/(360)*2\pi r=6\pi \\ \\ (\pi r)/(6)=6\pi \\ \\ r=6*6=36 \\ \text{The radius of the larger circle is 36}cm \end{gathered}`

The Circumferences:

- The circumferences of both circles are:

`\begin{gathered} Smaller\text{ circle:} \\ 2*\pi* r=2*\pi*18=36\pi \\ \\ \text{Larger circle:} \\ 2*\pi* r=2*\pi*36=72\pi \end{gathered}`

With these values, we can proceed to eliminate options.

Option 1:

This is WRONG because the radius of the Larger circle is 36cm, not 18cm

Option 2:

This is CORRECT because the radius of the smaller circle is 18cm

Option 3:

The length of segment NE is the difference in both radii which is 36cm - 18cm = 18cm. This is not equal to 9cm. Thus, this option is WRONG

Option 4:

This option is CORRECT because MR is also a difference in both radii and it is 18cm

Option 5:

This is CORRECT because the circumference of the larger circle is 72 pi

Option 6:

This is WRONG

Final Answer

The answers are:

Option 2

Option 4

Option 5