A and B are centres of two circles of radii 32 cm and 8 cm, respectively. CD is a direct common tangent to the circles. If the length of AB = 30 cm, then the length of CD will be:

The correct answer is 18 cm.

Given:

Radius of the first circle with centre A, ₁ = 32 cm

Radius of the second circle with centre B, ₂ = 8 cm

Distance between the centres A and B, = 30 cm

Formula:

The length of the direct common tangent () between two circles is given by the formula:

$\mathrm{CD}=\sqrt{d^2-{(r_1-r_2)}^2}$

Step-by-Step Derivation:

1. Find the difference between the two radii (₁ - ₂):

$r_1-r_2=32\text{cm}-8\text{cm}=24\text{cm}$

2. Substitute the values of = 30 cm and (₁ - ₂) = 24 cm into the formula:

$\mathrm{CD}=\sqrt{{30}^2-{24}^2}$

3. Calculate the squares inside the square root:

${30}^2=900$

${24}^2=576$

4. Subtract the values:

$\mathrm{CD}=\sqrt{900-576}=\sqrt{324}$

5. Take the square root of 324:

$\mathrm{CD}=18\text{cm}$

Thus, the length of the direct common tangent CD is 18 cm.