Explanation:

- Two circles touch each other externally at a point. The distance between their centers is the sum of their radii, which is 12 cm + 8 cm = 20 cm.

- When two circles touch externally, the length of the common external tangent can be found using the formula:

$$\text{Length of tangent}=\sqrt{(d^2-(r_1-r_2{)}^2)}$$

where $d$ is the distance between the centers ($20$ cm) and $r_1$ and $r_2$ are the radii of the circles ($12$ cm and $8$ cm).

- Substitute the values into the formula:

$$\text{Length of tangent}=\sqrt{({20}^2-(12-8{)}^2)}$$

$$=\sqrt{(400-16)}$$

$$=\sqrt{384}$$

$$=8\sqrt{6}$$

- The correct option representing the length of the tangent MN is therefore Option 2: $8\sqrt{6}$.

Correct Answer: Option 2: $8\sqrt{6}$

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