Question

Question 2 of 3Two concentric circles are shown. The length of the arc MN is 27 cm and the length of the arc RE is 47 cm.Which statements are correct? Select all that apply.The radius of the larger circle is 6 cm.The radius of the smaller circle is 6 cm.The length of the segment NE is 3 cm.The length of the segment MR is 6 cm.The circumference of the larger circle is 247 cm.The circumference of the larger circle is 12 cm.

Answer 1

The arc is only a part (fraction) cut out of the circumference

Formula:

`l_(arc)=(\Theta)/(360)2\pi r`

We see clearly that the arc is a fraction of the angle it subtends to the cicumference of the circle.

From here, we apply

`\begin{gathered} (\theta)/(360)=(l_(arc))/(C) \\ \text{Where C is the circumference of the circle} \end{gathered}`
`\begin{gathered} \text{For MN where l}_{arc\text{ = 2}\pi} \\ (\theta)/(360)=(l_(arc))/(C) \\ (60)/(360)=(2\pi)/(C) \\ C=\text{ }\frac{2\pi\text{ }*\text{ 360}}{60} \\ C=\text{ 12}\pi \\ \text{Calculating for radius} \\ C=\text{ 2}\pi r \\ 12\pi=\text{ 2}\pi r \\ \text{Dividing both sides by 2}\pi \\ r=\text{ 6}cm \end{gathered}`
`\begin{gathered} \text{For RE where l}_(arc)=\text{ 4}\pi \\ \text{The circumference of the circle will be twice the that of MN as its arc is twice the size} \\ \text{Therefore, the circumference C will be} \\ C=\text{ 2}*\text{ 12}\pi \\ C=\text{ 24}\pi \\ \text{Calculating for radius} \\ C=\text{ 2}\pi r \\ 24\pi=\text{ 2}\pi r \\ \text{Dividing both sides by 2}\pi \\ r=12cm \end{gathered}`
`\begin{gathered} segments\text{ MR and NE are the same distance} \\ \text{Hence, the answer is the difference betw}en\text{ the two radi}i \\ MR\text{ = NE = 12-6} \\ MR\text{ = NE = 6cm} \end{gathered}`

Final answers:

The radius of the smaller circle is 6 cm

The radius of the larger circle is 12 cm

The Circumference of the smaller circle is 12pi cm

The Circumference of the larger circle is 24pi cm

The length of the line segments MR and NE are 6cm each

Hence only Options B,D and E are correct