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. 

Answer 1

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.