A 6 cm long chord subtends an angle of 60° at the center of the circle. If the length of another chord of the same circle is 6/3 cm, then find the measure of the angle subtended by the chord at the center of the circle.

To find the angle subtended at the center of the circle by the chord of length \( \frac{6}{3} \, \text{cm} = 2 \, \text{cm} \), we can use the relationship between the length of a chord, the radius of the circle, and the angle subtended at the center.

### Step 1: Find the Radius of the Circle

Given that a chord of length \( 6 \, \text{cm} \) subtends an angle of \( 60^\circ \):

1. The formula for the length of a chord \( c \) in terms of the radius \( r \) and the angle \( \theta \) in radians is:

   \[
   c = 2r \sin\left(\frac{\theta}{2}\right)
   \]

2. Convert \( 60^\circ \) to radians:

   \[
   60^\circ = \frac{\pi}{3} \, \text{radians}
   \]

3. Substitute the values into the formula:

   \[
   6 = 2r \sin\left(\frac{\pi}{6}\right)
   \]

   Knowing that \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \):

   \[
   6 = 2r \cdot \frac{1}{2} \implies 6 = r
   \]

So, the radius \( r \) of the circle is \( 6 \, \text{cm} \).

### Step 2: Find the Angle for the 2 cm Chord

Now, we will find the angle \( \theta' \) subtended by the chord of length \( 2 \, \text{cm} \):

1. Use the chord length formula again:

   \[
   2 = 2r \sin\left(\frac{\theta'}{2}\right)
   \]

2. Substitute \( r = 6 \):

   \[
   2 = 2 \cdot 6 \sin\left(\frac{\theta'}{2}\right) \implies 2 = 12 \sin\left(\frac{\theta'}{2}\right)
   \]

3. Solve for \( \sin\left(\frac{\theta'}{2}\right) \):

   \[
   \sin\left(\frac{\theta'}{2}\right) = \frac{2}{12} = \frac{1}{6}
   \]

4. Find \( \frac{\theta'}{2} \):

   \[
   \frac{\theta'}{2} = \arcsin\left(\frac{1}{6}\right)
   \]

5. Therefore, \( \theta' \):

   \[
   \theta' = 2 \arcsin\left(\frac{1}{6}\right)
   \]

### Step 3: Calculate the Angle

Using a calculator:

\[
\theta' \approx 2 \cdot 0.167 \approx 0.334 \, \text{radians}
\]

To convert this back to degrees:

\[
\theta' \approx 0.334 \cdot \frac{180}{\pi} \approx 19.2^\circ
\]

### Conclusion

The angle subtended by the chord of length \( 2 \, \text{cm} \) at the center of the circle is approximately \( 19.2^\circ \).

Would you like more details or do you have any questions? Here are some related questions you might consider:

1. How would the angle change if the chord length was increased to 4 cm?
2. What happens to the angle if the radius of the circle is doubled?
3. Can you derive the chord length for an angle of \( 45^\circ \)?
4. How do you calculate the area of the sector formed by a chord?
5. What is the relationship between the radius and chord lengths for different angles? 

**Tip:** Always convert angles to radians when using trigonometric functions for consistency.

Explore how to find the angle subtended at the center of a circle by a chord. This example involves a 6 cm chord subtending a 60° angle, leading to the calculation of the angle for a 2 cm chord. Suitable for students in Grades 9-11.

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