Step 1: Understanding the Concept:
The relationship between the radius of a circle, the length of a tangent from an external point, and the distance from that point to the center of the circle forms a right-angled triangle. We can use this property to find the radius and then calculate the area.
Step 2: Key Formula or Approach:
Let:
$r$ be the radius of the circle.
$L$ be the length of the tangent from the external point.
$d$ be the distance from the external point to the center.
These three lengths form a right-angled triangle with $d$ as the hypotenuse. The relation is given by the Pythagorean theorem:
\[ r^2 + L^2 = d^2 \]
The area of a circle is given by $A = \pi r^2$.
Step 3: Detailed Explanation:
We are given:
Length of the tangent, $L = 8$ cm.
Distance from the center, $d = 11$ cm.
Using the Pythagorean relation:
\[ r^2 + 8^2 = 11^2 \]
\[ r^2 + 64 = 121 \]
Solve for $r^2$:
\[ r^2 = 121 - 64 \]
\[ r^2 = 57 \]
Now, calculate the area of the circle:
\[ A = \pi r^2 \]
\[ A = 57\pi \]
To find the numerical value, we can use the approximation $\pi \approx 3.14159$ or $\pi \approx 22/7$.
Using $\pi \approx 22/7$:
\[ A \approx 57 \times \frac{22}{7} = \frac{1254}{7} \approx 179.1428... \text{ cm}^2 \]
Using $\pi \approx 3.14$:
\[ A \approx 57 \times 3.14 = 178.98 \text{ cm}^2 \]
Both approximations are very close to the value in option (C).
Step 4: Final Answer:
The area of the circle is $57\pi$ cm$^2$, which is approximately 179.14 cm$^2$. Therefore, option (C) is correct.