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Two equal tangents PQ and PR are drawn from an external point P on a circle with centre O. if PO = 16√3m and the angle between the tangents is 60°. What is the length of each tangent and find the value of ∠QSR.

Two equal tangents PQ and PR are drawn from an external point P on a circle with centre O. if PO = 16√3m and the angle between the tangents is 60°. What is the length of each tangent and find the value of ∠QSR. Correct Answer 24m, 60°

Given:

PO = 16√3m

The angle between the tangents = 60°

Calculation:

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⇒ ∠QPO = ∠RPO

⇒ ∠QPO = ½ × ∠QPR = ½ × 60°

⇒ ∠QPO = 30°

In a Δ QPO,

⇒ ∠PQO = 90° (∵ PQ is a tangent)

⇒ cos 30° = PQ/PO

⇒ √3/2 = PQ/16√3

⇒ PQ = 24 m

⇒ PQ = PR = 24m

In a quadrilateral PQOR,

⇒ ∠PQO + ∠QOR + ∠ORP + ∠RPQ = 360°

⇒ 90° + ∠QOR + 90° + 60° = 360°

⇒ ∠QOR = 120°

⇒ ∠QOR = 2∠QSR (Same chord QR)

⇒ ∠QSR = 60°

∴ The length of each tangent is 24m and the value of ∠QSR is 60°.

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