Question

two tangents making an angle of 120 degrees are drawn to a circle of radius 6cm. What is the length of each tangent

Answer 1

*Refer to the attachment ☺

Given

Radius of the circle = 6cm

∠PAQ = 120°

Construction

Join AO

Solution

Since, the tangents are perpendicular to radius through the point of contact

∠OPA = 90° ----> (1)

∠OQA = 90° ----> (2)

In ∆OAP and ∆OAQ

AO = AO (Common)

AP = AQ (Length of the tangents to the same external point is equal)

∠OPA = ∠OQA {From (1) and (2)}

∴ ∆OAP ≡ ∆OAQ (by SAS Congreuency rule)

By CPCT, ∠OAP = ∠OAQ ----> (3)

∠OAP + ∠OAQ = ∠PAQ

=> ∠OAP + ∠OAP = 120° {From (3)}

=> 2 ∠OAP = 120°

=> ∠OAP = 120°/2 = 60°

Now, In ∆OAP, ∠P = 90°

Tan A = OP/AP

=> Tan 60° = 6/AP

=> √3 = 6/AP

=> AP = 6/√3

=> AP = 2√3 cm

∴ the length of each tangent = 2√3 cm