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In the given figure, AE and BC intersect each other at point D. If ∠CDE = 90°, AB = 5 cm, BD = 4 cm and CD = 9 cm, find AE.

In the figure, AE and BC intersect each other at point D. If ∠CDE = 90°, AB = 5 cm, BD = 4 cm and CD = 9 cm, find AE. Tangents and Intersecting Chords, Concise Mathematics Solutions ICSE Class 10.

Circles

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Answer

In right angle triangle ABD,

In the figure, AE and BC intersect each other at point D. If ∠CDE = 90°, AB = 5 cm, BD = 4 cm and CD = 9 cm, find AE. Tangents and Intersecting Chords, Concise Mathematics Solutions ICSE Class 10.

⇒ AB2 = BD2 + AD2

⇒ 52 = 42 + AD2

⇒ 25 = 16 + AD2

⇒ AD2 = 25 - 16

⇒ AD2 = 9

⇒ AD = 9\sqrt{9}

⇒ AD = 3 cm.

We know that,

If two chords of circle intersect internally or externally then the product of the lengths of their segment is equal.

⇒ AD × DE = CD × BD

⇒ 3 × DE = 9 × 4

⇒ DE = 9×43\dfrac{9 \times 4}{3}

⇒ DE = 12 cm.

From figure,

AE = AD + DE = 3 + 12 = 15 cm.

Hence, AE = 15 cm.

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