In fig 3.28 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.

In fig 3.28 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.

Figure: The circles touch each other externally at E (A–E–B); both circles lie on the same side of line \(l\).

Doubtly Asked 24 Oct 2026 Save WhatsApp Report

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Answer: CD = \(4\sqrt{6}\) cm

AB = 4 + 6 = 10 cm (circles touching externally). AC ⊥ \(l\) and BD ⊥ \(l\) (tangent ⊥ radius), so AC ∥ BD and □ACDB is a trapezium with AC = 4, BD = 6.

Draw AM ⊥ BD with M on BD. Then □ACDM is a rectangle, so AM = CD and MD = AC = 4, giving BM = 6 − 4 = 2 cm.

In right-angled \(\triangle AMB\):

\[AM^2 = AB^2 - BM^2 = 100 - 4 = 96\] \[CD = AM = \sqrt{96} = 4\sqrt{6}\ \text{cm}\]

Doubtly AI · 24 Oct 2026 AI-verified · Oct 2026 Report

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In fig 3.28 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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