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A tangent AB is drawn from an exterior point A that intersects the circle at point B. Another line AD is drawn from A, which intersects the circle at points C and D, such that CD = 5 cm. If the sum of the lengths of tangent AB and line AD is 15 cm, then find the length of the line AD.

A tangent AB is drawn from an exterior point A that intersects the circle at point B. Another line AD is drawn from A, which intersects the circle at points C and D, such that CD = 5 cm. If the sum of the lengths of tangent AB and line AD is 15 cm, then find the length of the line AD. Correct Answer 9 cm

Let the length of line AD be 'x' cm

⇒ Length of tangent AB = (15 – x) cm

Line AD is a secant to the circle as it intersects the circle at two points C and D,

As we know, when a secant and a tangent are drawn to a circle from the same exterior point, then the square of the length of the tangent is equal to the product of the length of the external segment of the secant and the total length of the secant

⇒ AB2 = AC × AD      ---- (1)

Given, CD = 5 cm,

⇒ AC = AD – CD = (x – 5) cm

Substituting in (1), we get,

⇒ (15 – x)2 = (x – 5) × x

⇒ 225 + x2 – 30x = x2 – 5x

⇒ 25x = 225

⇒ x = 225/25 = 9 cm

∴ Length of line AD = 9 cm

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