Question

Suppose there are two circles where the radius of one circle is twice the radius of the other circle. Each circle has an arc where the measures of the corresponding central angles are the same. What is the relationship between the arc lengths in the two circles? Explain. (2 points)​

Answer 1

`L=2l` where
`l,L` denote arc lengths of two circles

Explanation:

Let
`l,L` denote arc lengths of two circles,
`r,R` denote corresponding radii and

`\alpha _1\,,\alpha _2` denote the corresponding central angles.

So,

`l=r\alpha _1` and
`L=R\alpha _2`

This implies
`\alpha _1=(l)/(r)` and
`\alpha _2=(L)/(R)`

As each circle has an arc where the measures of the corresponding central angles are the same,
`\alpha _1=\alpha _2`

`(l)/(r)=(L)/(R)`

As radius of one circle is twice the radius of the other circle,

`R=2r`

`(l)/(r)=(L)/(2r)\\(l)/(1) =(L)/(2)\\L=2l`