Two circles of radii 8 cm and 2 cm respectively touch each other externally at the point A. PQ is the direct common tangent of these two circles of centres `O_(1) and O_(2)` respectively. The length of PQ is equal to :

Options

A. 4 cm
B. 8 cm
C. 2 cm
D. 3 cm

Answer

To find the length of the direct common tangent (PQ) between two circles that touch each other externally, we can use the formula for the length of the direct common tangent:

\[ PQ = \sqrt{d^2 - (R_1 - R_2)^2} \]

Where:
- \(d\) is the distance between the centers of the two circles,
- \(R_1\) is the radius of the first circle,
- \(R_2\) is the radius of the second circle.

### Step-by-Step Solution:

1. **Identify the radii of the circles**:
- Radius of the first circle \(R_1 = 8 \, \text{cm}\)
- Radius of the second circle \(R_2 = 2 \, \text{cm}\)

2. **Calculate the distance between the centers of the circles**:
Since the circles touch each other externally, the distance \(d\) between the centers \(O_1\) and \(O_2\) is given by:
\[ d = R_1 + R_2 = 8 \, \text{cm} + 2 \, \text{cm} = 10 \, \text{cm} \]

3. **Calculate the difference of the radii**:
\[ R_1 - R_2 = 8 \, \text{cm} - 2 \, \text{cm} = 6 \, \text{cm} \]

4. **Substitute the values into the tangent length formula**:
\[ PQ = \sqrt{d^2 - (R_1 - R_2)^2} \]
\[ PQ = \sqrt{10^2 - 6^2} \]

5. **Calculate \(d^2\) and \((R_1 - R_2)^2\)**:
\[ d^2 = 10^2 = 100 \]
\[ (R_1 - R_2)^2 = 6^2 = 36 \]

6. **Subtract the squares**:
\[ PQ = \sqrt{100 - 36} = \sqrt{64} \]

7. **Calculate the square root**:
\[ PQ = 8 \, \text{cm} \]

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