Let's solve this step by step.

1. Calculate the central angle subtended by the chord:

Given:

• Chord length, $AB = 6 \, \text{cm}$
• Radius of the circle, $r = 7 \, \text{cm}$

To find the angle $\theta$ subtended by the chord at the center of the circle, we use the following formula for the central angle in terms of the chord length and radius:

$\cos\left(\frac{\theta}{2}\right) = \frac{\text{half of the chord length}}{\text{radius}}$

$\cos\left(\frac{\theta}{2}\right) = \frac{6/2}{7} = \frac{3}{7}$

Thus,

$\frac{\theta}{2} = \cos^{-1}\left(\frac{3}{7}\right)$

Let's calculate $\theta$.

2. Find the length of the minor arc:

The length of the minor arc $L$ is given by the formula:

$L = r \cdot \theta$

Where $\theta$ is in radians and $r = 7 \, \text{cm}$.

3. Calculate the area of the smaller region (segment):

The area of the segment (smaller region) can be found by subtracting the area of the isosceles triangle formed by the chord and the center from the area of the sector.

• The area of the sector is:

$A_{\text{sector}} = \frac{1}{2} r^2 \theta$

• The area of the triangle is:

$A_{\text{triangle}} = \frac{1}{2} r^2 \sin\theta$

Thus, the area of the segment is:

$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}$

Let's calculate these values!### Results:

1. Length of the minor arc: $15.79 \, \text{cm}$
2. Area of the smaller region (segment): $36.29 \, \text{cm}^2$
3. Central angle subtended by the chord: $2.256 \, \text{radians}$

These are the required values for the minor arc length and the area of the smaller region inside the circle.

Do you want further clarification or details on any step?

Here are some additional related questions you might find useful:

1. How do you calculate the area of the major segment?
2. How does the arc length formula change for a different radius?
3. Can this method be used with degrees instead of radians?
4. What if the chord was longer than the radius?
5. How do you calculate the area of a full circle sector?

Tip: When using radians, remember that $2\pi$ radians is equivalent to a full circle or $360^\circ$.