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 those two circles of centres `O_1` and `O_2` respectively. Then length of PQ is equal to

Options

A. 2 cm
B. 3 cm
C. 5 cm
D. 8 cm

Answer

To find the length of the direct common tangent PQ between two circles with radii \( R_1 = 8 \) cm and \( R_2 = 2 \) cm, we can use the formula for the length of the direct common tangent between two circles:

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

### Step-by-Step Solution:

1. **Identify the radii of the circles:**
- Let \( R_1 = 8 \) cm (radius of the first circle).
- Let \( R_2 = 2 \) cm (radius of the second circle).

2. **Use the formula for the length of the direct common tangent:**
- The formula is given by:
\[ PQ = \sqrt{(R_1 + R_2)^2 - (R_1 - R_2)^2} \]

3. **Calculate \( R_1 + R_2 \) and \( R_1 - R_2 \):**
- \( R_1 + R_2 = 8 + 2 = 10 \)
- \( R_1 - R_2 = 8 - 2 = 6 \)

4. **Substitute these values into the formula:**
- Now we calculate:
\[ PQ = \sqrt{10^2 - 6^2} \]
- Calculate \( 10^2 = 100 \) and \( 6^2 = 36 \).

5. **Perform the subtraction:**
- \( 100 - 36 = 64 \)

6. **Take the square root:**
- \( PQ = \sqrt{64} = 8 \) cm

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

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