To calculate the area of the shaded segment in the given circle, we need to break down the problem into a few steps:

1. Identify the Circle's Parameters:

   • Radius $r = 8 \, \text{cm}$.
   • Chord $AB = 14 \, \text{cm}$.

2. Calculate the Angle Subtended by the Chord $AB$ at the Circle's Center: Since the chord length and the radius are known, we can use the cosine rule in triangle $OAB$ (where $O$ is the center) to find the angle $\theta$ subtended by $AB$ at the center.

   In $\triangle OAB$: $AB^2 = 2r^2(1 - \cos \theta)$ Substituting values: $14^2 = 2 \times 8^2 (1 - \cos \theta)$ $196 = 128 (1 - \cos \theta)$ $\cos \theta = 1 - \frac{196}{128}$ $\cos \theta = \frac{32}{128} = 0.25$ So, $\theta = \cos^{-1}(0.25) \approx 75.52^\circ$ (or $1.318$ radians).

3. Calculate the Area of the Sector $OAB$: The area of a sector of a circle is given by: $\text{Area of sector} = \frac{1}{2} r^2 \theta$ Substituting $r = 8 \, \text{cm}$ and $\theta = 1.318 \, \text{radians}$: $\text{Area of sector} = \frac{1}{2} \times 8^2 \times 1.318$ $= \frac{1}{2} \times 64 \times 1.318$ $\approx 42.176 \, \text{cm}^2$

4. Calculate the Area of Triangle $OAB$: We can use the formula for the area of a triangle with two sides and the included angle: $\text{Area of } \triangle OAB = \frac{1}{2} r^2 \sin \theta$ Using $r = 8 \, \text{cm}$ and $\sin 1.318 \approx 0.968$: $\text{Area of } \triangle OAB = \frac{1}{2} \times 8 \times 8 \times 0.968$ $= \frac{1}{2} \times 64 \times 0.968$ $\approx 30.976 \, \text{cm}^2$

5. Calculate the Area of the Shaded Segment: The area of the shaded segment is the area of the sector minus the area of the triangle: $\text{Area of shaded segment} = \text{Area of sector} - \text{Area of } \triangle OAB$ $\approx 42.176 - 30.976$ $\approx 11.2 \, \text{cm}^2$

Final Answer:

The area of the shaded segment is approximately 11.2 cm².

Would you like more details on any step or have questions?

Here are some related questions you might explore:

1. How is the cosine rule derived, and when is it used?
2. What are other methods to find the area of a circular segment?
3. How would the solution change if the radius or chord length were different?
4. Can you apply the sine rule to find angles in circular segments?
5. How does converting between radians and degrees affect calculations?

Tip: Always ensure your calculator is set to the correct mode (degrees or radians) based on the angle format in your problem.