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Question
In the given figure, the circles with centres A and B touch each other at E. Line l is a common tangent which touches the circles at C and D respectively. Find the length of seg CD if the radii of the circles are 4 cm, 6 cm. 
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Solution
If two circles touch each other externally, then the distance between their centres is equal to the sum of their radii.
∴ AB = AE + EB = 4 cm + 6 cm = 10 cm
It is given that l is a common tangent which touches the circles at C and D.
∴ ∠ACD = ∠CDF = 90º ...(Tangent theorem)
Draw AF ⊥ BD.
Here, ACDF is a rectangle.
∴ CD = AF and DF = AC ...(Opposite sides of the rectangle are equal)
FB = BD − DF
FB = 6 cm − 4 cm
= 2 cm
In right ∆AFB,
\[{AB}^2 = {AF}^2 + {FB}^2 \]
∴ 102 = AF2 + 22
∴ 100 = AF2 + 4
∴ AF2 = 100 − 4
∴ AF2 = 96
∴ AF = `sqrt96`
∴ AF = `sqrt(16 × 6)`
∴ CD = AF = \[4\sqrt{6}\] cm
Thus, the length of seg CD is \[4\sqrt{6}\] cm.
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