The centres of two circles are 36 cm apart. If the radii of these two circles are 15 cm and 9 cm, respectively, then what is the sum of the lengths (in cm) of a direct common tangent and a transverse common tangent of these two circles?

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Given:
- Distance between centers (d) = 36 cm
- Radii: r1 = 15 cm, r2 = 9 cm

We need to find the sum of the lengths of the direct common tangent and the transverse common tangent.

Formulas:
1. Length of direct common tangent (L1) = √(d² - (r1 - r2)²)
2. Length of transverse common tangent (L2) = √(d² - (r1 + r2)²)

Step 1: Calculate (r1 - r2) and (r1 + r2)
- r1 - r2 = 15 - 9 = 6
- r1 + r2 = 15 + 9 = 24

Step 2: Calculate L1
L1 = √(36² - 6²) = √(1296 - 36) = √1260

Step 3: Calculate L2
L2 = √(36² - 24²) = √(1296 - 576) = √720

Step 4: Simplify the square roots
- √1260 = √(36 × 35) = 6√35
- √720 = √(36 × 20) = 6√20

Step 5: Sum of lengths
L1 + L2 = 6√35 + 6√20 = 6(√35 + √20)

Step 6: Simplify √35 and √20
- √35 = √(5 × 7) = √5 × √7
- √20 = √(4 × 5) = 2√5

So,
L1 + L2 = 6(√5√7 + 2√5) = 6√5(√7 + 2)

Therefore, the sum of the lengths of the direct and transverse common tangents is 6√5(√7 + 2) cm.

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