The distance between the centres of equal circles each of radius 3 cm is 10 cm. The length of a transverse tangent is:

Options

A. 4 cm
B. 6 cm
C. 8 cm
D. 10 cm

Answer

To find the length of the transverse tangent between two equal circles, we can use the formula:

\[ L = \sqrt{d^2 - (r_1 + r_2)^2} \]

where:
- \(L\) is the length of the transverse tangent,
- \(d\) is the distance between the centers of the circles,
- \(r_1\) and \(r_2\) are the radii of the circles.

### Step-by-Step Solution:

1. **Identify the values**:
- The radius of each circle \(r_1 = r_2 = 3 \, \text{cm}\).
- The distance between the centers of the circles \(d = 10 \, \text{cm}\).

2. **Calculate the sum of the radii**:
\[ r_1 + r_2 = 3 + 3 = 6 \, \text{cm} \]

3. **Square the distance \(d\)**:
\[ d^2 = 10^2 = 100 \]

4. **Square the sum of the radii**:
\[ (r_1 + r_2)^2 = 6^2 = 36 \]

5. **Subtract the square of the sum of the radii from the square of the distance**:
\[ d^2 - (r_1 + r_2)^2 = 100 - 36 = 64 \]

6. **Take the square root to find the length of the transverse tangent**:
\[ L = \sqrt{64} = 8 \, \text{cm} \]

### Final Answer:
The length of the transverse tangent is \(8 \, \text{cm}\).

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