A chord of the larger among two concentric circles is of length 10 cm and it is tangent to the smaller circle. What is the area (in cm ) of the annular portion between the two circles?

Options

A. `(5pi)/2`
B. `25 pi`
C. `5pi`
D. `10pi`

Answer

To solve the problem, we need to find the area of the annular portion between two concentric circles given that a chord of the larger circle is tangent to the smaller circle.

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have two concentric circles: a larger circle with radius \( R \) and a smaller circle with radius \( r \).
- A chord of the larger circle has a length of 10 cm and is tangent to the smaller circle.

2. **Identify the Chord's Properties**:
- Since the chord is tangent to the smaller circle, the distance from the center of the circles to the chord is equal to the radius of the smaller circle \( r \).
- The chord divides into two equal segments, so each segment (half the chord) is \( \frac{10}{2} = 5 \) cm.

3. **Apply the Pythagorean Theorem**:
- In the right triangle formed by the radius of the larger circle (hypotenuse), the distance from the center to the chord (one leg), and half the chord length (the other leg), we can write:
\[ R^2 = r^2 + 5^2 \]
- This simplifies to:
\[ R^2 = r^2 + 25 \]

4. **Find the Area of the Annular Portion**:
- The area of the annular portion between the two circles is given by the difference in their areas:
\[ \text{Area} = \pi R^2 - \pi r^2 \]
- Factoring out \( \pi \):
\[ \text{Area} = \pi (R^2 - r^2) \]
- Substituting the expression for \( R^2 - r^2 \) from step 3:
\[ R^2 - r^2 = 25 \]
- Thus, the area becomes:
\[ \text{Area} = \pi \times 25 = 25\pi \text{ cm}^2 \]

### Final Answer:
The area of the annular portion between the two circles is \( 25\pi \) cm².