If a chord of a circle of radius 5 cm is a tangent to another circle of radius 3 cm, both the circles being concentric, then the length of the chord is

Options

A. 10 cm
B. 12.5 cm
C. 8 cm
D. 7 cm

Answer

To solve the problem, we need to find the length of the chord of a circle with a radius of 5 cm, which is tangent to another concentric circle with a radius of 3 cm.

### Step-by-Step Solution:

1. **Understand the Geometry**:
We have two concentric circles. The larger circle has a radius of 5 cm, and the smaller circle has a radius of 3 cm. The chord of the larger circle is tangent to the smaller circle.

2. **Draw the Diagram**:
Draw two concentric circles. Label the center as O. The larger circle (radius 5 cm) and the smaller circle (radius 3 cm) should be clearly marked.

3. **Identify the Tangent Point**:
Let the point where the chord touches the smaller circle be point T. Since the chord is tangent to the smaller circle, the distance from the center O to the point T is equal to the radius of the smaller circle, which is 3 cm.

4. **Draw the Perpendicular from the Center to the Chord**:
Draw a perpendicular line from the center O to the chord. Let the point where this perpendicular meets the chord be point M. This line OM bisects the chord.

5. **Use the Pythagorean Theorem**:
In the right triangle OMT:
- OM is the distance from the center to the chord (which is 3 cm).
- OT is the radius of the larger circle (which is 5 cm).
- MT is half the length of the chord (let's denote it as x).

According to the Pythagorean theorem:
\[ OT^2 = OM^2 + MT^2 \]
Substituting the known values:
\[ 5^2 = 3^2 + x^2 \]
\[ 25 = 9 + x^2 \]
\[ x^2 = 25 - 9 \]
\[ x^2 = 16 \]
\[ x = 4 \]

6. **Calculate the Length of the Chord**:
Since MT is half the length of the chord, the full length of the chord (let's denote it as L) is:
\[ L = 2 \times MT = 2 \times 4 = 8 \text{ cm} \]

### Final Answer:
The length of the chord is **8 cm**.