Given:
Circle C and circle R are similar.
The length of arc AB is [tex]s = \frac{22 \pi}{9}[/tex]
The radius of circle C (AC) = 4 unit
The radius of circle R (QR) =6 unit
To find the length of arc QP.
Formula
The relation between s, r and [tex]\theta[/tex] is
[tex]arclength = 2\pi r \frac{\theta}{360}[/tex]
where,
s be the length of the arc
r be the radius
[tex]\theta[/tex] be the angle.
Now,
For circle C
Taking r = 4
According to the problem,
[tex]2 \pi r \frac{\theta}{360} = \frac{22 \pi}{9}[/tex]
or, [tex]2r \frac{\theta}{360} = \frac{22}{9}[/tex] [ eliminating [tex]\theta[/tex] from both side]
or, [tex]\theta = \frac{(22)(360)}{(9)(2)(4)}[/tex]
or, [tex]\theta = 110^\circ[/tex]
Again,
For circle R
Taking, r = 6 and [tex]\theta = 110^\circ[/tex] we get,
The length of arc QP is
[tex]arc length = 2\pi (6)(\frac{110}{360} )[/tex]
or, [tex]arclength = \frac{11 \pi}{3}[/tex]
Hence,
The length of QP is [tex]\frac{11 \pi}{3}[/tex]. Option C.
