Two circles with centre O and A touch each other externally. The radius of the first circle with centre O is 6 cm. The radius of the second circle with centre A is 3 cm. Find the length of their common tangent CB (in centimetres).

The correct answer is Option 1: $6\sqrt{2}$ cm.

Given Data:

- Two circles touch each other externally.

- Radius of the first circle with centre O, $R_1=6\text{cm}$.

- Radius of the second circle with centre A, $R_2=3\text{cm}$.

- CB is a direct common tangent to both circles.

Step-by-Step Explanation:

1. Distance between the centers (d):

Since the two circles touch each other externally, the distance between their centers O and A is equal to the sum of their radii:

$d=OA=R_1+R_2=6+3=9\text{cm}$

2. Formula for the length of the direct common tangent (CB):

The length of a direct common tangent to two circles is given by the standard formula:

$CB=\sqrt{d^2-{(R_1-R_2)}^2}$

3. Substituting the values into the formula:

$CB=\sqrt{9^2-{(6-3)}^2}$

$CB=\sqrt{81-3^2}$

$CB=\sqrt{81-9}$

$CB=\sqrt{72}$

4. Simplifying the radical:

$CB=\sqrt{36\times 2}=6\sqrt{2}\text{cm}$

Thus, the length of the common tangent CB is $6\sqrt{2}$ cm.