Question

Circle 1 has a central angle
θ
and a radius of 9 inches. The angle intercepts an arc length of 6 inches.
Circle 2 has the same central angle
θ
as Circle 1, but a radius of 3 inches.
image
What is the arc length of Circle 2?
A.
2 in

B.
3 in

C.
6 in

D.
9 in

Answer 1

To find the arc length of Circle 2, we can use the relationship between arc length, radius, and central angle. The formula for arc length \( L \) is given by:

\[
L = r \theta
\]

where \( r \) is the radius and \( \theta \) is the central angle in radians.

From Circle 1, we know that:

- The radius \( r_1 = 9 \) inches.
- The arc length \( L_1 = 6 \) inches.

Using the formula for Circle 1, we can solve for \( \theta \):

\[
6 = 9 \theta
\]

To find \( \theta \):

\[
\theta = \frac{6}{9} = \frac{2}{3} \text{ radians}
\]

Now we can use this value of \( \theta \) to find the arc length of Circle 2, which has a radius \( r_2 = 3 \) inches.

Using the same formula for Circle 2:

\[
L_2 = r_2 \theta = 3 \cdot \frac{2}{3} = 2 \text{ inches}
\]

Therefore, the arc length of Circle 2 is:

\[
\boxed{2 \text{ in}}
\]