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 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 = ∠BDC = 90º (Tangent at any point of a circle is perpendicular to the radius throught the point of contact)

Draw AF ⊥ BD.



Here, ACDF is a rectangle.

∴ CD = AF and DF = AC (Opposite sides of rectangle are equal)

BF = BD − DF = BD − AC = 6 cm − 4 cm = 2 cm

In right ∆ABF,

$$\begin{gathered} {\mathrm{AB}}^2={\mathrm{AF}}^2+{\mathrm{BF}}^2 \\ \Rightarrow \mathrm{AF}=\sqrt{{\mathrm{AB}}^2-{\mathrm{BF}}^2} \\ \Rightarrow \mathrm{AF}=\sqrt{{10}^2-2^2} \\ \Rightarrow \mathrm{AF}=\sqrt{100-4}=\sqrt{96}=4\sqrt{6}\mathrm{cm} \end{gathered}$$
∴ CD = AF = $4\sqrt{6}$ cm

Thus, the length of seg CD is $4\sqrt{6}$ cm.